http://www.map.mpim-bonn.mpg.de/api.php?action=feedcontributions&user=Marek+Kaluba&feedformat=atomManifold Atlas - User contributions [en]2022-09-24T22:17:48ZUser contributionsMediaWiki 1.18.4http://www.map.mpim-bonn.mpg.de/Talk:Inertia_group_I_(Ex)Talk:Inertia group I (Ex)2013-08-30T12:56:01Z<p>Marek Kaluba: solution</p>
<hr />
<div><wikitex>;<br />
First observe, that instead of forming a connected sum $M\#\Sigma_f$ we may think of it as cutting a disc $M-D^n=M^\bullet$ and glueing the disc back identifying boundary spheres via diffeomorphism $f$. By the theorem of Cerf definition of $M^\bullet$ does not depend on the embedding $D^n \hookrightarrow M$.<br />
<br />
If $\Sigma\in I(M)$ and $G\colon M\#\Sigma\to M$ is a diffeomorphism, find a decomposition $M\#\Sigma=M^\bullet\cup_f D^n$ and set $F=G|_{M^\bullet}$.<br />
<br />
Conversely suppose that there exist a diffeomorphism $F\colon M^\bullet\to M^\bullet$ such that $F|_{\partial(M^\bullet)=S^{n-1}}=f$. Then glue a disc via identity to the source and target $M^\bullet$ and extend $F$ as identity. Composition of inclusion and $F$, $S^{n-1}\to M^\bullet\to M^\bullet$ is equal to $f$, hence (by definition) the target is equal to $M\#\Sigma$. This proves that $\Sigma \in I(M)$.<br />
<br />
''(Is the Cerfs thm properly applied?)''<br />
</wikitex></div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Talk:Structure_set_(Ex)Talk:Structure set (Ex)2013-08-29T12:14:00Z<p>Marek Kaluba: Added solution for h-structure set</p>
<hr />
<div><wikitex>;<br />
'''Solution'''(simple structure set):<br />
<br />
We begin with the map $\mathcal{S}^s(M) \to \mathcal{M}^s(M)$, from the simple structure set of manifolds simply homotopy equivalent to the orbit space and then we show two things:<br />
* If manifolds $N$ and $N'$ simply homotopy equivalent to $M$ are diffeomorphic then their images by the map belong to the same orbit of $\mathcal{E}^s(M)$-action on $\mathcal{S}^s(M)$.<br />
* If two elements of $\mathcal{S}^s(M)$ belong to the same orbit of $\mathcal{E}^s(M)$-action, then they are diffeomorphic.<br />
<br />
Let $N$ be a smooth manifold and $f\colon N\to M$ a simple homotopy equivalence. Consider a map which takes $N$ to $[(N,f)]\in \mathcal{S}^s(M)$. Suppose now that $N'$ is a manifold diffeomorphic to $N$, and $f'\colon N'\to M$ a simple homotopy equivalence (possibly $N'=N$ and $f\nsim f'$). Then there exists $h\colon M\to M$ such that the following diagram commutes.<br />
<br />
$$<br />
\xymatrix{<br />
N \ar[r]^{f} \ar[d]^d_{\cong} & M \ar@{.>}[d]^{h}\\<br />
N'\ar[r]^{f'}& M } <br />
$$<br />
<br />
Map $h$ is given by composition $f'\circ d\circ f^{-1}$ (the homotopy inverse) and hence is a simple homotopy equivalence. The commutativity of the diagram tells us that in $\mathcal{S}^s(M)$ we have the following equalities. <br />
$$h\cdot [(N,f)] = [(N,h\circ f)]=[(N,f'\circ d\circ f^{-1}\circ f\simeq f'\circ d)]=[(N',f')],$$ <br />
where $\cdot$ denotes the $\mathcal{E}^s(M)$-action. Therefore $[(N,f)]$ and $[(N',f')]$ belong to the same orbit.<br />
<br />
Suppose now, that $[(N,f)],[(N',f')]\in \mathcal{S}^s(M)$ belong to the same orbit of $\mathcal{E}^s(M)$-action. It means, that there exist a simple homotopy equivalence $h\colon M\to M$ such that $$h\cdot[(N,f)]=[(N,h\circ f)]=[(N',f')].$$ But equality in the simple structure set $\mathcal{S}^s(M)$ is just the existence of a diffeomorphism $d\colon N\to N'$ making the following diagram commute.<br />
<br />
$$<br />
\xymatrix{<br />
N \ar[r]^{f} \ar@{.>}[d]^d & M \ar[d]^{h}\\<br />
N'\ar[r]^{f'}& M } <br />
.$$<br />
<br />
'''Solution''' (homotopy structure set):<br />
<br />
Basically, the proof follows the same line. However in this case equality $[(N_0,f_0)]=[(N_1,f_1)]$ in $\mathcal{S}(M)$ (by definition) is existence of an $h$-cobordism $$(H,\partial_0H,\partial_0H)\colon (W,\partial_0W,\partial1W)\to (M\times I,M\times\{0\},M\times\{1\})$$ satisfying $f_i=\partial_iH\circ g_i\colon N_i\to \partial_iH\to M\times \{i\}$ for some orientation preserving diffeomorphisms $g_i$, $i=0,1$. But the h-cobordism gives us a homotopy equivalence $d\colon N_0\to N_1$ and the same formula for $h\colon M\to M$ applies.<br />
<br />
Analogously, equality $h\cdot [(N,f)] = \cdot [(N',f')]$ in $\mathcal{S}^h(M)$ implies existence of an $h$-cobordism between $N$ and $N'$ extending $h\circ f$ and $f'$.<br />
<br />
</wikitex></div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Talk:Structure_set_(Ex)Talk:Structure set (Ex)2013-08-28T07:03:37Z<p>Marek Kaluba: </p>
<hr />
<div><wikitex>;<br />
'''Solution''':<br />
<br />
We begin with the map $\mathcal{S}^s(M) \to \mathcal{M}^s(M)$, from the simple structure set of manifolds simply homotopy equivalent to the orbit space and then we show two things:<br />
* If manifolds $N$ and $N'$ simply homotopy equivalent to $M$ are diffeomorphic then their images by the map belong to the same orbit of $\mathcal{E}^s(M)$-action on $\mathcal{S}^s(M)$.<br />
* If two elements of $\mathcal{S}^s(M)$ belong to the same orbit of $\mathcal{E}^s(M)$-action, then they are diffeomorphic.<br />
<br />
Let $N$ be a smooth manifold and $f\colon N\to M$ a simple homotopy equivalence. Consider a map which takes $N$ to $[(N,f)]\in \mathcal{S}^s(M)$. Suppose now that $N'$ is a manifold diffeomorphic to $N$, and $f'\colon N'\to M$ a simple homotopy equivalence (possibly $N'=N$ and $f\nsim f'$). Then there exists $h\colon M\to M$ such that the following diagram commutes.<br />
<br />
$$<br />
\xymatrix{<br />
N \ar[r]^{f} \ar[d]^d_{\cong} & M \ar@{.>}[d]^{h}\\<br />
N'\ar[r]^{f'}& M } <br />
$$<br />
<br />
Map $h$ is given by composition $f'\circ d\circ f^{-1}$ (the homotopy inverse) and hence is a simple homotopy equivalence. The commutativity of the diagram tells us that in $\mathcal{S}^s(M)$ we have the following equalities. <br />
$$h\cdot [(N,f)] = [(N,h\circ f)]=[(N,f'\circ d\circ f^{-1}\circ f\simeq f'\circ d)]=[(N',f')],$$ <br />
where $\cdot$ denotes the $\mathcal{E}^s(M)$-action. Therefore $[(N,f)]$ and $[(N',f')]$ belong to the same orbit.<br />
<br />
Suppose now, that $[(N,f)],[(N',f')]\in \mathcal{S}^s(M)$ belong to the same orbit of $\mathcal{E}^s(M)$-action. It means, that there exist a simple homotopy equivalence $h\colon M\to M$ such that $$h\cdot[(N,f)]=[(N,h\circ f)]=[(N',f')].$$ But equality in the simple structure set $\mathcal{S}^s(M)$ is just the existence of a diffeomorphism $d\colon N\to N'$ making the following diagram commute.<br />
<br />
$$<br />
\xymatrix{<br />
N \ar[r]^{f} \ar@{.>}[d]^d & M \ar[d]^{h}\\<br />
N'\ar[r]^{f'}& M } <br />
.$$<br />
<br />
<br />
<br />
</wikitex></div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Talk:Structure_set_(Ex)Talk:Structure set (Ex)2013-08-28T06:57:25Z<p>Marek Kaluba: typos</p>
<hr />
<div><wikitex>;<br />
'''Solution''':<br />
<br />
We begin with the map $\mathcal{S}^s(M) \to \mathcal{M}(M)$, from the simple structure set of manifolds simply homotopy equivalent to the orbit space and then we show two things:<br />
* If manifolds $N$ and $N'$ simply homotopy equivalent to $M$ are diffeomorphic then their images by the map belong to the same orbit of $\mathcal{E}^s(M)$-action on $\mathcal{S}^s(M)$.<br />
* If two elements of $\mathcal{S}^s(M)$ belong to the same orbit of $\mathcal{E}^s(M)$-action, then they are diffeomorphic.<br />
<br />
Let $N$ be a smooth manifold and $f\colon N\to M$ a simple homotopy equivalence. Consider a map which takes $N$ to $[(N,f)]\in \mathcal{S}^s(M)$. Suppose now that $N'$ is a manifold diffeomorphic to $N$, and $f'\colon N'\to M$ a simple homotopy equivalence (possibly $N'=N$ and $f\nsim f'$). Then there exists $h\colon M\to M$ such that the following diagram commutes.<br />
<br />
$$<br />
\xymatrix{<br />
N \ar[r]^{f} \ar[d]^d_{\cong} & M \ar@{.>}[d]^{h}\\<br />
N'\ar[r]^{f'}& M } <br />
$$<br />
<br />
Map $h$ is given by composition $f'\circ d\circ f^{-1}$ (the homotopy inverse) and hence is a simple homotopy equivalence. The commutativity of the diagram tells us that in $\mathcal{S}^s(M)$ we have the following equalities. <br />
$$h\cdot [(N,f)] = [(N,h\circ f)]=[(N,f'\circ d\circ f^{-1}\circ f\simeq f'\circ d)]=[(N',f')],$$ <br />
where $\cdot$ denotes the $\mathcal{E}^s(M)$-action. Therefore $[(N,f)]$ and $[(N',f')]$ belong to the same orbit.<br />
<br />
Suppose now, that $[(N,f)],[(N',f')]\in \mathcal{S}^s(M)$ belong to the same orbit of $\mathcal{E}^s(M)$-action. It means, that there exist a simple homotopy equivalence $h\colon M\to M$ such that $$h\cdot[(N,f)]=[(N,h\circ f)]=[(N',f')].$$ But equality in the simple structure set $\mathcal{S}^s(M)$ is just the existence of a diffeomorphism $d\colon N\to N'$ making the following diagram commute.<br />
<br />
$$<br />
\xymatrix{<br />
N \ar[r]^{f} \ar@{.>}[d]^d & M \ar[d]^{h}\\<br />
N'\ar[r]^{f'}& M } <br />
.$$<br />
<br />
<br />
<br />
</wikitex></div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Talk:Structure_set_(Ex)Talk:Structure set (Ex)2013-08-27T22:59:03Z<p>Marek Kaluba: Solution to simple h. eq. version</p>
<hr />
<div><wikitex>;<br />
'''Solution''':<br />
<br />
We begin with a map from the set of manifolds simply homotopy equivalent to $M$ to $\mathcal{S}^s(M)$ and then we show two things:<br />
* if manifolds $N$ and $N'$ simply homotopy equivalent to $M$ are diffeomorphic then their images by the map belong to the same orbit of $\mathcal{E}^s(M)$-action on $\mathcal{S}^s(M)$.<br />
* if two elements of $\mathcal{S}^s(M)$ belong to the same orbit of $\mathcal{E}^s(M)$-action, then we will show that they are diffeomorphic.<br />
<br />
Let $N$ be a smooth manifold and $f\colon N\to M$ a simple homotopy equivalence. Consider a map which takes $N$ to $[(N,f)]\in \mathcal{S}^c(M)$. Suppose now that $N'$ is a manifold diffeomorphic to $N$, and $f'\colon N'\to M$ a simple homotopy equivalence (possibly $N'=N$ and $f\nsim f'$). Then there exists $h\colon M\to M$ such that the following diagram commutes.<br />
<br />
$$<br />
\xymatrix{<br />
N \ar[r]^{f} \ar[d]^d_{\cong} & M \ar@{.>}[d]^{h}\\<br />
N'\ar[r]^{f'}& M } <br />
$$<br />
<br />
Map $h$ is given by composition $f'\circ d\circ f^{-1}$ (the homotopy inverse) and hence is a simple homotopy equivalence. The commutativity of the diagram tells us that in $\mathcal{S}^s(M)$ we have the following equalities. <br />
$$h\cdot [(N,f)] = [(N,h\circ f)]=[(N,f'\circ d\circ f^{-1}\circ f\simeq f'\circ d)]=[(N',f')],$$ <br />
where $\cdot$ denotes the $\mathcal{E}^s(M)$-action. Therefore $[(N,f)]$ and $[(N',f')]$ belong to the same orbit.<br />
<br />
Suppose now, that $[(N,f)],[(N',f')]\in \mathcal{S}^s(M)$ belong to the same orbit of $\mathcal{E}^s(M)$-action. It means, that there exist a simple homotopy equivalence $h\colon M\to M$ such that $$h\cdot[(N,f)]=[(N,h\circ f)]=[(N',f')].$$ But equality in the simple structure set $\mathcal{S}^s(M)$ is just the existence of a diffeomorphism $d\colon N\to N'$ making the following diagram commute.<br />
<br />
$$<br />
\xymatrix{<br />
N \ar[r]^{f} \ar@{.>}[d]^d & M \ar[d]^{h}\\<br />
N'\ar[r]^{f'}& M } <br />
.$$<br />
<br />
<br />
<br />
</wikitex></div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Structure_set_(Ex)Structure set (Ex)2013-08-27T16:07:34Z<p>Marek Kaluba: Corrected formulation of the exercise; Added the simple version</p>
<hr />
<div><wikitex>;<br />
The exercise has two parts depending on whether we are talking about homotopy equivalences or ''simple'' homotopy equivalence. Both proofs are rather careful reading of definitions and share the same idea.<br />
<br />
<br />
* Let $\mathcal{S}(M) = \{[(N,f) \text{ such that } f\colon N\simeq M]\}$ be the structure set of a closed manifold and let $\mathcal{E}(M)$ be the group of homotopy self-equivalences of $M$. Note that $\mathcal{E}(M)$ acts on $\mathcal{S}(M)$ by post composition: $$ \begin{array}{rcl} \mathcal{S}(M) \times \mathcal{E}(M) & \to & \mathcal{S}(M),\\ ([f:N\to M],[g]) &\mapsto& [g\circ f: N\to M].\end{array}$$ Show that the set $\mathcal{M}(M):= \mathcal{S}(M)/\mathcal{E}(M)$ is in bijection with the set of h-cobordism classes of manifolds homotopy equivalent to $M$.<br />
<br />
* Let $\mathcal{S}^s(M) = \{[(N,f) \text{ such that } f\colon N\simeq_s M]\}$ be the simple structure set of a closed manifold and let $\mathcal{E}^s(M)$ be the group of simple homotopy self-equivalences of $M$. Note that $\mathcal{E}^s(M)$ acts on $\mathcal{S}^s(M)$ by post composition: $$ \begin{array}{rcl} \mathcal{S}^s(M) \times \mathcal{E}^s(M) & \to & \mathcal{S}^s(M),\\ ([f:N\to M],[g]) &\mapsto & [g\circ f: N\to M].\end{array}$$ Show that the set $\mathcal{M}^s(M):= \mathcal{S}^s(M)/\mathcal{E}^s(M)$ is in bijection with the set of diffeomorphism classes of manifolds homotopy equivalent to $M$.<br />
<br />
</wikitex><br />
== References ==<br />
{{#RefList:}}<br />
[[Category:Exercises]]<br />
[[Category:Exercises without solution]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/5-manifolds_with_fundamental_group_of_order_25-manifolds with fundamental group of order 22013-02-07T00:36:24Z<p>Marek Kaluba: /* Introduction */</p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
The classification of simply-connected 5-manifolds was achieved by Smale \cite{Smale} and Barden \cite{Barden} in the 1960s. The surgery and modified surgery theories provide a toolkit to attack the classification problem of high dimensional manifolds. Dimension 5 is the lowest dimension where the theories apply (except for the 4-dimensional TOP-category), therefore an understanding of the classification of 5-manifolds beyond the Smale-Barden results is expectable. On the other hand, it's known (\cite{Markov}, \cite{Kervaire}) that every finitely generated group can be realized as the fundamental group of a manifold of dimension $\ge$ 4. Therefore, a practical approach towards the classification of 5-manifolds is to fix a fundamental group in advance and consider the classification of manifolds with the given fundamental group. <br />
<br />
From this point of view, the fist step one might take is the group $\Zz_2$, which is the simplest nontrivial group since it has the least number of elements. (Of course if we take the point of view of generators and relations the rank 1 free group $\Zz$ is the simplest one.) <br />
<br />
Any other point one should take into account concerning the classification of manifolds with nontrivial fundamental groups $\pi_1$ is that the higher homotopy groups $\pi_i$ ($i \ge 2$) are modules over the group ring $\Zz[\pi_1]$, which are apparently homotopy invariants. Especially when we consider $5$-manifolds $M^5$, the $\Zz[\pi_1]$-module structure of $\pi_2(M)$ will play an important role in the classification. And it's not a surprise that the first clear classification result obtained is the trivial module case. <br />
<br />
Most part of this item will be a survey of the classification result of 5-manifolds $M^5$ with fundamental group $\pi_1(M)=\Zz_2$, $\pi_2(M)$ torsion free and is a trivial $\Zz[\Zz_2]$-module obtained in \cite{Hambleton&Su} .<br />
<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<wikitex>;<br />
First some examples known from other context. <br />
*$S^2 \times \mathbb R P^3$;<br />
* $X^5(q)$ $q=1,3,5,7$ are the 5-dimensional fake real projective spaces with $X(1)=\mathbb R \mathrm P^5$ (There are exactly 4 in the smooth category). The meaning of $q$ will be clear in the section ``Invariants".<br />
</wikitex><br />
<br />
<br />
=== Circle bundles over simply-connected 4-manifolds ===<br />
<wikitex>;<br />
<br />
Let $X^4$ be a closed simply-connected topological 4-manifold, $\xi$ be a complex line bundle over $X$ with Chern class $c_1(\xi) \in H^2(X;\Zz)$. Let the divisibility of $c_1(\xi)$ be $k$ (i.e. $c_1(\xi)$ is $k$ multiple of a primitive element in $H^2(X;\Zz)$, then the sphere bundle $S(\xi) = \colon M$ is a 5-manifold with fundament group $\Zz_k$ and $\pi_2(M)$ is a free abelian group of rank $\mathrm{rank} H_2(X)-1$, and $\pi_2(M)$ is a trivial module over the group ring. A priori $M$ is a topological manifold. The smoothing problem is addressed by the following<br />
{{beginthm|Proposition}}\cite[Proposition 4.2]{Hambleton&Su}<br />
Assume $\xi$ is nontrivial. If $k$ is odd, then $M$ admits a smooth structure; if $k$ is even, then $M$ admits a smooth structure if and only if the Kirby-Siebenmann invariant of $X$ is $0$. <br />
{{endthm}}<br />
* $k=1$, $M$ is a simply-connected 5-manifold. The identification of $M$ with manifolds in the standard list of simply-connected 5-manifolds given by the Smale-Barden classification was done in \cite{Duan&Liang}.<br />
* $k=2$, we have a class of orientable 5-manifolds with fundamental group $\Zz_2$, $\pi_2(M)$ a free abelian group, and a trivial module over the group ring. The classification of these manifolds was the motivation of \cite{Hambleton&Su}.<br />
</wikitex><br />
<br />
=== Connected sum along $S^1$ ===<br />
<wikitex>;<br />
In the Smale-Barden's list of simply-connected 5-manifolds, manfolds are constructed from simple building blocks by the connected sum operation. In the world of manifolds with fundamental group $\Zz_2$, the connected sum operation is not closed. The ``connected sum along $S^1$" operation $\sharp_{S^1}$ will do the job. <br />
{{beginthm|Definition}}<br />
Let $M_1^5$, $M_2^5$ be two oriented 5-manifolds with $\pi_1=\Zz_2$. Let $E_i \subset M_i$ be the normal bundle of an embedded $S^1$ in $M_i$ representing the nontrivial element in the fundamental group. $E_i$ is a rank 4 trivial vector bundle over $S^1$. Choose trivialisations of $E_1$ and $E_2$, and identify the disk bundles of $E_1$ and $E_2$ using the chosen identification (such that the identification is orientation-reversing, with respect to the induced orientations on the disk bundles from the ambient manifolds), we obtain a new manifold denoted by $M_1 \sharp_{S^1} M_2$. <br />
{{endthm}}<br />
<br />
Notice that $\sharp_{S^1}$ is not well-defined, the ambiguity comes from the identification of the two normal bundles, measured by $\pi_1SO(4) = \Zz_2$. To eliminate the ambiguity we need more structures on the tangent bundles of $M_i$ and require that $\sharp_{S^1}$ preserves the structures. (Analogous to the connected sum situation where orientations on manifolds are needed.) This will be explained in more detail in the next section.<br />
<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<!-- == Acknowledgments ==<br />
...<br />
<br />
== Footnotes ==<br />
<references/> --><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- == External links ==<br />
* The Wikipedia page about [[Wikipedia:Page_name|link text]]. --><br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]<br />
[[Catefory:5-Manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Petrie_conjecturePetrie conjecture2012-10-24T23:12:12Z<p>Marek Kaluba: /* Problem */</p>
<hr />
<div>{{Stub}}<br />
== Problem ==<br />
<wikitex>;<br />
If a compact Lie group $G$ acts smoothly and non-trivially on a closed smooth manifold $M$, what constraints does this place on the topology of $M$ in general and on the Pontrjagin classes of $M$ in particular?<br />
In the case where $M$ is [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]], $M \simeq \CP^n$, Petrie {{cite|Petrie1972}} restricted his attention to actions of the Lie group <br />
$S^1 = \{z \in \mathbb{C} \ \colon \ |z| = 1\}$ and proved that if the fixed point set of the action consists only of isolated fixed points, then the Pontrjagin classes of $M$ are determined by the representations of $S^1$ at the fixed points. Motivated by this result, Petrie {{cite|Petrie1972}} posed the following conjecture.<br />
<br />
{{beginthm|Conjecture|(Petrie conjecture)}} <br />
Suppose that $S^1$ acts smoothly and non-trivially on a closed smooth $2n$-manifold $M \simeq \CP^n$. Then the total Pontrjagin class $p(M)$ of $M$ agrees with that of $\CP^n$, i.e., <br />
$p(M) = (1+x^2)^{n}$ for a generator $x$ of $H^2(M; \mathbb{Z})$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
== Progress to date ==<br />
<wikitex>;<br />
As of December 21, 2010, the Petrie conjecture has not been confirmed in general. However, it has been proven in the following special cases.<br />
* Petrie {{cite|Petrie1973}} has verified his conjecture under the assumption that the manifold $M \simeq \CP^n$ admits a smooth action of the torus $T^n$.<br />
* By the work of {{cite|Dejter1976}}, the Petrie conjecture is true if $\dim M = 6$, i.e., $M \simeq \CP^3$ and hence, if $\dim M \leq 6$. <br />
* Related results go back to {{cite|Musin1978}} and {{cite|Musin1980}}, in particular, the latter work shows that the Petrie conjecture holds if $\dim M = 8$, i.e., $M \simeq \CP^4$. <br />
* According to {{cite|Hattori1978}}, the Petrie conjecture holds if $M$ admits an invariant almost complex structure with the first Chern class $c_1(M) = (n+1)x$.<br />
* Other special cases where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1975/76}}, {{cite|Iberkleid1978}}.<br />
* By {{cite|Tsukada&Washiyama1979}} and {{cite|Masuda1981}}, the Petrie conjecture is true if the fixed point set consists of three or four connected components.<br />
* Masuda {{cite|Masuda1983}} proved the Petrie conjecture in the case where $M$ admits a specific smooth action of $T^k$ for $k \geq 2$. <br />
* The work of {{cite|James1985}} confirms the result of {{cite|Musin1980}} that the Petrie conjecture is true if $\dim M = 8$, i.e., $M \simeq \CP^4$. <br />
* According to {{cite|Dessai2002}}, the Petrie conjecture holds if $M$ admits an appropriate smooth action of $Pin(2)$ and $\dim M \leq 22$. <br />
* It follows from {{cite|Dessai&Wilking2004}} that the Petrie conjecture holds if $M$ admits a smooth action of $T^k$ and $\dim M \leq 8k-4$.<br />
</wikitex><br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
Masuda and Suh {{cite|Masuda&Suh2008}} posed the following question about the invariance of Pontrjagin classes for toric $2n$-manifolds.<br />
{{beginthm|Question}}<br />
For two toric $2n$-manifolds with isomorphic cohomology rings, is it true that any isomorphism between the cohomology rings preserves the Pontrjagin classes of the two manifolds?<br />
{{endthm}}<br />
<br />
A symplectic version of the Petrie conjecture is discussed by Tolman {{cite|Tolman2010}}. In particular, the following question has been posed.<br />
{{beginthm|Question}} <br />
If the circle $S^1$ acts in a Hamiltonian way on a compact symplectic manifold $M$ with $H^{2i}(M;\Rr) \cong H^{2i}(\CP^n; \Rr)$ for all $i \geq 0$,<br />
is it true that $H^{j}(M;\Zz) \cong H^{j}(\CP^n; \Zz)$ for all $j \geq 0$? Is the total Chern class of $M$ determined by the cohomology ring $H^*(M;\Zz)$? <br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<!-- --><br />
[[Category:Problems]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Oberwolfach_Surgery_Seminar_2012:_ExercisesOberwolfach Surgery Seminar 2012: Exercises2012-05-30T10:14:46Z<p>Marek Kaluba: </p>
<hr />
<div>This page lists the exercises for consideration during the Blockseminar. Participants are encouraged to work on the solution page of each and to contribute to the discussion pages of both exercises and solutions.<br />
<br />
* [[Oberwolfach Surgery Seminar 2012: General information|General information]]<br />
<!-- * [[Oberwolfach Surgery Seminar 2012: Program|Program]]--><br />
* [[Oberwolfach Surgery Seminar 2012: Glossary|Glossary]]<br />
* [[:Category:Surgery|Surgery on the Manifold Atlas]]<br />
<br />
== Talks 1-4 == <br />
# [[Tangent bundles of bundles (Ex)]] - [[User:Suyang|Yang Su]]/ [[User: Suzhixu|Zhixu Su]]<br />
# [[Microbundles (Ex)]] - [[User:Marek Kaluba|Marek Kaluba]]<br />
# [[Fibre homotopy trivial bundles (Ex)]] - [[User:Daniel Kasprowski|Daniel Kasprowski]]<br />
# [[Normal bundles in products of spheres (Ex)]]<br />
# [[Non-reducible Spivak Normal Fibrations (Ex)]]<br />
# [[Reducible Poincaré Complexes (Ex)]]<br />
# [[S-duality I (Ex)]] - [[User:Martin Palmer|Martin Palmer]]<br />
# [[S-duality II (Ex)]] - [[User:Patrickorson|Patrick Orson]]<br />
<br />
== Talks 5-7 ==<br />
# [[Kernel formation (Ex)]] - [[User:Marek Kaluba|Marek Kaluba]]<br />
# [[Presentations (Ex)]]<br />
# [[Quadratic formations (Ex)]] <br />
# [[Wall realisation (Ex)]]<br />
# [[Topological structures on products of spheres (Ex)]]<br />
# [[Surgery obstruction map I (Ex)]]<br />
# [[Structures on M x I (Ex)]]<br />
<br />
== Talks 8-10 ==<br />
# [[Quadratic forms I (Ex)]]<br />
# [[Forms and chain complexes I (Ex)]]<br />
# [[Forms and chain complexes II (Ex)]]<br />
# [[Formations and chain complexes I]]<br />
# [[Formations and chain complexes II (Ex)]]<br />
# [[Novikov additivity I (Ex)]]<br />
<br />
== Talks 11-13 ==<br />
# [[Chain duality I (Ex)]]<br />
# [[Chain duality II (Ex)]]<br />
# [[Chain duality III (Ex)]]<br />
# [[Chain duality IV (Ex)]]<br />
# [[Chain duality V (Ex)]]<br />
# [[Chain duality VI (Ex)]]<br />
# [[Chain duality VII (Ex)]]<br />
# [[Supplement I (Ex)]]<br />
# [[Supplement II (Ex)]]<br />
# [[Supplement III (Ex)]]<br />
<br />
== Talks 14&15 ==<br />
<br />
# [[Novikov additivity II (Ex)]]<br />
# [[Algebraic surgery X (Ex)]]<br />
<br />
== Talks 16-18==<br />
*<br />
[[Category:Oberwolfach Surgery Seminar 2012]]<br />
[[Category:Exercises]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Oberwolfach_Surgery_Seminar_2012:_ExercisesOberwolfach Surgery Seminar 2012: Exercises2012-05-30T09:50:29Z<p>Marek Kaluba: </p>
<hr />
<div>This page lists the exercises for consideration during the Blockseminar. Participants are encouraged to work on the solution page of each and to contribute to the discussion pages of both exercises and solutions.<br />
<br />
* [[Oberwolfach Surgery Seminar 2012: General information|General information]]<br />
<!-- * [[Oberwolfach Surgery Seminar 2012: Program|Program]]--><br />
* [[Oberwolfach Surgery Seminar 2012: Glossary|Glossary]]<br />
* [[:Category:Surgery|Surgery on the Manifold Atlas]]<br />
<br />
== Talks 1-4 == <br />
# [[Tangent bundles of bundles (Ex)]] - [[User:Suyang|Yang Su]]/ [[User: Suzhixu|Zhixu Su]]<br />
# [[Microbundles (Ex)]] - - [[User:Marek Kaluba|Marek Kaluba]]<br />
# [[Fibre homotopy trivial bundles (Ex)]] - [[User:Daniel Kasprowski|Daniel Kasprowski]]<br />
# [[Normal bundles in products of spheres (Ex)]]<br />
# [[Non-reducible Spivak Normal Fibrations (Ex)]]<br />
# [[Reducible Poincaré Complexes (Ex)]]<br />
# [[S-duality I (Ex)]] - [[User:Martin Palmer|Martin Palmer]]<br />
# [[S-duality II (Ex)]]<br />
<br />
== Talks 5-7 ==<br />
# [[Kernel formation (Ex)]]<br />
# [[Presentations (Ex)]]<br />
# [[Quadratic formations (Ex)]] <br />
# [[Wall realisation (Ex)]]<br />
# [[Topological structures on products of spheres (Ex)]]<br />
# [[Surgery obstruction map I (Ex)]]<br />
# [[Structures on M x I (Ex)]]<br />
<br />
== Talks 8-10 ==<br />
# [[Quadratic forms I (Ex)]]<br />
# [[Forms and chain complexes I (Ex)]]<br />
# [[Forms and chain complexes II (Ex)]]<br />
# [[Formations and chain complexes I]]<br />
# [[Formations and chain complexes II (Ex)]]<br />
# [[Novikov additivity I (Ex)]]<br />
<br />
== Talks 11-13 ==<br />
# [[Chain duality I (Ex)]]<br />
# [[Chain duality II (Ex)]]<br />
# [[Chain duality III (Ex)]]<br />
# [[Chain duality IV (Ex)]]<br />
# [[Chain duality V (Ex)]]<br />
# [[Chain duality VI (Ex)]]<br />
# [[Chain duality VII (Ex)]]<br />
# [[Supplement I (Ex)]]<br />
# [[Supplement II (Ex)]]<br />
# [[Supplement III (Ex)]]<br />
<br />
== Talks 14&15 ==<br />
<br />
# [[Novikov additivity II (Ex)]]<br />
# [[Algebraic surgery X (Ex)]]<br />
<br />
== Talks 16-18==<br />
*<br />
[[Category:Oberwolfach Surgery Seminar 2012]]<br />
[[Category:Exercises]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/MicrobundleMicrobundle2012-05-30T09:49:02Z<p>Marek Kaluba: </p>
<hr />
<div>== Introduction ==<br />
<wikitex>;<br />
The concept of Microbundle of dimension $n$ was first introduced in {{cite|Milnor1964}} to give a model for the tangent bundle of an n-dimensional [[topological manifold]]. Later \cite{Kister1964} showed that every microbundle uniquely determines a topological $\Rr^n$-bundle.<br />
<br />
{{beginthm|Definition|{{cite|Milnor1964}} }}<br />
An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ such that there is a sequence $$B\xrightarrow{i} E\xrightarrow{j} B$$ and the following conditions hold.<br />
#$j\circ i=\id_B$<br />
#for all $x\in B$ there exist open neigbourhood $U\subset B$, an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism $$h \colon V \to U\times \mathbb{R}^n$$<br />
which makes the following diagram commute:<br />
$$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}} $$<br />
{{endthm}}<br />
<br />
{{beginrem|Example|{{citeD|Milnor1964|Lemma 2.1}}}}<br />
Let $M$ be topological $n$-manifold, let $\Delta_M \colon M \to M \times M$ be the diagonal map and let $p_1 \colon M \times M \to M$ be the projection onto the first factor. Then<br />
$$ (M \times M, M, \Delta_M, p_1) $$<br />
is an $n$-dimensional microbundle.<br />
{{endrem}}<br />
{{beginrem|Example}} Let $\pi \colon E \to B$ be a topological $\Rr^n$-bundle with zero section $s \colon B \to E$. Then $$(E, B, s, \pi)$$ <br />
is an $n$-dimensional microbundle.<br />
{{endrem}}<br />
<br />
{{beginthm|Definition}}<br />
Two microbundles $(E_n,B,i_n,j_n)$, $n=1,2$ over the same space $B$ are isomorphic if there exist neighbourhoods $V_1\subset E_1$ of $i_1(B)$ and $V_2\subset E_2$ of $i_2(B)$ and a homeomorphism $H\colon V_1\to V_2$ making the following diagram commute. <br />
<br />
$$<br />
\xymatrix{<br />
& V_1 \ar[dd]^H\ar[rd]^{j_1|_{V_1}} \\<br />
B\ar[ru]^{i_1}\ar[rd]_{i_2} & & B \\<br />
& V_2 \ar[ru]_{j_2|_{V_2}}<br />
}<br />
$$<br />
<br />
{{endthm|Definition}}<br />
<br />
{{beginthm|Theorem|\cite{Kister1964|Theorem 2} }}<br />
Let $(E, B, i, j)$ be an $n$-dimensional microbundle. Then there is a neighbourhood of $i(B)$, $E_1 \subset E$ such that:<br />
# $E_1$ is the total space of a topological $\Rr^n$-bundle over $B$.<br />
# The inclusion $E_1 \to E$ is a microbundle isomorphism<br />
# If $E_2 \subset E$ is any other such neighbourhood of $i(B)$ then there is a $\Rr^n$-bundle isomorphism $(E_1 \to B) \cong (E_2 \to B)$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/MicrobundleMicrobundle2012-05-30T09:47:55Z<p>Marek Kaluba: </p>
<hr />
<div>== Introduction ==<br />
<wikitex>;<br />
The concept of Microbundle of dimension $n$ was first introduced in {{cite|Milnor1964}} to give a model for the tangent bundle of an n-dimensional [[topological manifold]]. Later \cite{Kister1964} showed that every microbundle uniquely determines a topological $\Rr^n$-bundle.<br />
<br />
{{beginthm|Definition|{{cite|Milnor1964}} }}<br />
An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ such that there is a sequence $$B\xrightarrow{i} E\xrightarrow{j} B$$ and the following conditions hold.<br />
#$j\circ i=\id_B$<br />
#for all $x\in B$ there exist open neigbourhood $U\subset B$, an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism $$h \colon V \to U\times \mathbb{R}^n$$<br />
which makes the following diagram commute:<br />
$$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd] \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}} $$<br />
{{endthm}}<br />
<br />
{{beginrem|Example|{{citeD|Milnor1964|Lemma 2.1}}}}<br />
Let $M$ be topological $n$-manifold, let $\Delta_M \colon M \to M \times M$ be the diagonal map and let $p_1 \colon M \times M \to M$ be the projection onto the first factor. Then<br />
$$ (M \times M, M, \Delta_M, p_1) $$<br />
is an $n$-dimensional microbundle.<br />
{{endrem}}<br />
{{beginrem|Example}} Let $\pi \colon E \to B$ be a topological $\Rr^n$-bundle with zero section $s \colon B \to E$. Then $$(E, B, s, \pi)$$ <br />
is an $n$-dimensional microbundle.<br />
{{endrem}}<br />
<br />
{{beginthm|Definition}}<br />
Two microbundles $(E_n,B,i_n,j_n)$, $n=1,2$ over the same space $B$ are isomorphic if there exist neighbourhoods $V_1\subset E_1$ of $i_1(B)$ and $V_2\subset E_2$ of $i_2(B)$ and a homeomorphism $H\colon V_1\to V_2$ making the following diagram commute. <br />
<br />
$$<br />
\xymatrix{<br />
& V_1 \ar[dd]^H\ar[rd]^{j_1|_{V_1}} \\<br />
B\ar[ru]^{i_1}\ar[rd]_{i_2} & & B \\<br />
& V_2 \ar[ru]_{j_2|_{V_2}}<br />
}<br />
$$<br />
<br />
{{endthm|Definition}}<br />
<br />
{{beginthm|Theorem|\cite{Kister1964|Theorem 2} }}<br />
Let $(E, B, i, j)$ be an $n$-dimensional microbundle. Then there is a neighbourhood of $i(B)$, $E_1 \subset E$ such that:<br />
# $E_1$ is the total space of a topological $\Rr^n$-bundle over $B$.<br />
# The inclusion $E_1 \to E$ is a microbundle isomorphism<br />
# If $E_2 \subset E$ is any other such neighbourhood of $i(B)$ then there is a $\Rr^n$-bundle isomorphism $(E_1 \to B) \cong (E_2 \to B)$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Microbundles_(Ex)Microbundles (Ex)2012-05-30T09:42:04Z<p>Marek Kaluba: </p>
<hr />
<div><wikitex>;<br />
{{beginthm|Exercise|{{citeD|Milnor1964|Theorem 2.2}}}}<br />
# Let $M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle.<br />
{{endthm}}<br />
<br />
{{beginthm|Exercise|{{citeD|Milnor1964|Theorem 2.2}}}}<br />
# Let $M$ be a smooth manifold. Show that $TM$ and $\xi_M$ are isomorphic microbundles.<br />
{{endthm}}<br />
<br />
</wikitex><br />
== References ==<br />
{{#RefList:}}<br />
[[Category:Exercises]]<br />
[[Category:Exercises with solution]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Microbundles_(Ex)Microbundles (Ex)2012-05-30T08:16:51Z<p>Marek Kaluba: moved to with solution category</p>
<hr />
<div><wikitex>;<br />
{{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1, Theorem 2.2}}}}<br />
# Let $M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle.<br />
# Let $M$ be a smooth manifold. Show that $TM$ and $\xi_M$ are isomorphic microbundles.<br />
{{endthm}}<br />
</wikitex><br />
== References ==<br />
{{#RefList:}}<br />
[[Category:Exercises]]<br />
[[Category:Exercises with solution]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Talk:Microbundles_(Ex)Talk:Microbundles (Ex)2012-05-30T08:16:10Z<p>Marek Kaluba: 2nd exercise finisher</p>
<hr />
<div><wikitex>;<br />
First, You should get familiar with the definition of [[Microbundle|microbundle]].<br />
<br />
{{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1}}}}<br />
Let $M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle.<br />
{{endthm}}<br />
{{beginproof}} <br />
<br />
Let $M$ be a topological manifold. Then the composition $p_1\circ\Delta_M$ sends $x\mapsto (x,x)\mapsto x$, so the first condition in the definition is satisfied.<br />
<br />
To prove that the second condition is satisfied we need to use local chart around $x$.<br />
Choose $U$ to be one of the open sets coming from atlas of $M$ and let $\phi\colon U\to \mathbb{R}^n$ be associated chart. The obvious choice for neighbourhood $V\subset M\times M$ is to take $U\times U$. The first naive candidate for $h\colon V=U\times U\to U\times\mathbb{R}^n$ would be map $\id\times \phi$. However such $h$ fails to make the following diagram commute<br />
<br />
$$<br />
\xymatrix{<br />
&U\times U\ar[rd]^{p_1}\ar[dd]^h&\\<br />
U\ar[ru]^{\Delta_M}\ar[rd]_{\id\times\{0\}} & & U\\<br />
&U\times \Rr^n\ar[ru]_{p_1}&} <br />
$$<br />
since $(u,u)$ is mapped to $(u,\phi(u))$ and $\phi(u)$ doesn't necessarily be $0$ (well, it depends on our definition of an atlas of a manifold?). We need just a small adjustment: $h(u,v)=(u,h(u)-h(v))$.<br />
{{endproof}}<br />
<br />
<br />
{{beginthm|Exercise|{{citeD|Milnor1964|Theorem 2.2}}}}<br />
Let $M$ be a (paracompact!) smooth manifold. Show that $TM$ and $\xi_M$ are isomorphic microbundles.<br />
{{endthm}}<br />
{{beginproof}}<br />
We have two concurring definitions of micro-tangent bundle. The first is given by the exercise above, the second by forgeting about the vector bundle structure on $TM$ and treating it just as a microbundle $(TM, M, \pi,s_0)$ where $M\xrightarrow{s_0} TM$ is the zero section.<br />
<br />
To fix the notation please consult the definition of microbundle isomorphism on page on [[Microbundle|microbundles]] .<br />
<br />
In our case we have <br />
$$<br />
\xymatrix{<br />
& V\ar[dd]^H \ar[rd]^{\pi}&\\<br />
M\ar[dr]_{\Delta_M}\ar[ur]^{s_0} & & M\\<br />
& M\times M \ar[ru]_{p_1}&}<br />
$$<br />
<br />
where $V\subset TM$ is an open neighbourhood of the zero section.<br />
<br />
We need to find a neighbourhood $V$ and a map $H\colon V\to U\times U$ such that points in the zero section ($\{(x,0)\}$ in local coordinates) are mapped to the diagonal $\{(x,x)\}$. <br />
<br />
At each point this is easy: Fix $b\in M$ and let $V'\subset TM$ be a neighbourhood of $i(b)$ coming from the vector bundle structure. Choose a trivialization $V'\to M\times \Rr^n$ and then set $H\colon M\times \Rr^n\to M\times M$, $$H(x,v)=(x,\exp(b,v)).$$By definition of $\exp$ we have $H(b,0)=(b,\exp(b,0))=(b,b)$.<br />
<br />
However, we may now let $b$ vary as $x$ does and define $$H(x,v)=(x,\exp(x,v)).$$As checked before this map maps the zero section to the diagonal. By definition of the expotential map the derivative of $H$ is non-vanishing along the zero section, so by the inverse function theorem there exist a neighbourhood $V\subset TM$ of $M$ on which $H$ is a diffeomorphism.<br />
{{endproof}}<br />
<br />
</wikitex></div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Talk:Microbundles_(Ex)Talk:Microbundles (Ex)2012-05-30T00:10:02Z<p>Marek Kaluba: 2nd exercise WIP</p>
<hr />
<div><wikitex>;<br />
In case of daubts You should get familiar with the definition of [[Microbundle|microbundle]].<br />
<br />
{{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1}}}}<br />
Let $M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle.<br />
{{endthm}}<br />
{{beginproof}} <br />
<br />
Let $M$ be a topological manifold. Then the composition $p_1\circ\Delta_M$ sends $x\mapsto (x,x)\mapsto x$, so the first condition in the definition is satisfied.<br />
<br />
To prove that the second condition is satisfied we need to use local chart around $x$.<br />
Choose $U$ to be one of the open sets coming from atlas of $M$ and let $\phi\colon U\to \mathbb{R}^n$ be associated chart. The obvious choice for neighbourhood $V\subset M\times M$ is to take $U\times U$. The first naive candidate for $h\colon V=U\times U\to U\times\mathbb{R}^n$ would be map $\id\times \phi$. However such $h$ fails to make the following diagram commute<br />
<br />
$$<br />
\xymatrix{<br />
&U\times U\ar[rd]^{p_1}\ar[dd]^h&\\<br />
U\ar[ru]^{\Delta_M}\ar[rd]_{\id\times\{0\}} & & U\\<br />
&U\times \Rr^n\ar[ru]_{p_1}&} <br />
$$<br />
since $(u,u)$ is mapped to $(u,\phi(u))$ and $\phi(u)$ doesn't necessarily be $0$ (well, it depends on our definition of an atlas of a manifold?). We need just a small adjustment: $h(u,v)=(u,h(u)-h(v))$.<br />
{{endproof}}<br />
<br />
<br />
{{beginthm|Exercise|{{citeD|Milnor1964|Theorem 2.2}}}}<br />
Let $M$ be a (paracompact!) smooth manifold. Show that $TM$ and $\xi_M$ are isomorphic microbundles.<br />
{{endthm}}<br />
{{beginproof}}<br />
We have two concurring definitions of micro-tangent bundle. The first is given by the exercise above, the second by forgeting about the vector bundle structure on $TM$ and treating it just as a microbundle $(TM, M, \pi,s_0)$ where $M\xrightarrow{s_0} TM$ is the zero section.<br />
<br />
To fix the notation please consult the definition of microbundle isomorphism on page on [[Microbundle|microbundles]] .<br />
<br />
In our case we have <br />
$$<br />
\xymatrix{<br />
& V\ar[dd]^H \ar[rd]^{p_1}&\\<br />
M\ar[dr]_{\Delta_M}\ar[ur]^{s_0} & & M\\<br />
& M\times M \ar[ru]&}<br />
$$<br />
<br />
where $V\subset TM$ is an open neighbourhood of the zero section.<br />
{{endproof}}<br />
<br />
We need to find a map $H\colon V\to <br />
</wikitex></div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/MicrobundleMicrobundle2012-05-29T20:37:26Z<p>Marek Kaluba: Definition of isomorphism</p>
<hr />
<div>== Introduction ==<br />
<wikitex>;<br />
The concept of Microbundle of dimension $n$ was first introduced in {{cite|Milnor1964}} to give a model for the tangent bundle of an n-dimensional [[topological manifold]]. Later \cite{Kister1964} showed that every microbundle uniquely determines a topological $\Rr^n$-bundle.<br />
<br />
{{beginthm|Definition|{{cite|Milnor1964}} }}<br />
An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ such that there is a sequence $$B\xrightarrow{i} E\xrightarrow{j} B$$ and the following conditions hold.<br />
#$j\circ i=\id_B$<br />
#for all $x\in B$ there exist open neigbourhood $U\subset B$, an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism $$h \colon V \to U\times \mathbb{R}^n$$<br />
which makes the following diagram commute:<br />
$$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd] \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}} $$<br />
{{endthm}}<br />
<br />
{{beginrem|Example|{{citeD|Milnor1964|Lemma 2.1}}}}<br />
Let $M$ be topological $n$-manifold, let $\Delta_M \colon M \to M \times M$ be the diagonal map and let $p_1 \colon M \times M \to M$ be the projection onto the first factor. Then<br />
$$ (M \times M, M, \Delta_M, p_1) $$<br />
is an $n$-dimensional microbundle.<br />
{{endrem}}<br />
{{beginrem|Example}} Let $\pi \colon E \to B$ be a topological $\Rr^n$-bundle with zero section $s \colon B \to E$. Then $$(E, B, s, \pi)$$ <br />
is an $n$-dimensional microbundle.<br />
{{endrem}}<br />
<br />
{{beginthm|Definition}}<br />
Two microbundles $(E_n,X,i_n,j_n)$, $n=1,2$ over the same space $X$ are isomorphic if there exist neighbourhoods $V_1\subset E_1$ of $i_1(B)$ and $V_2\subset E_2$ of $i_2(B)$ and a homeomorphism $H\colon V_1\to V_2$ making the following diagram commute. <br />
<br />
$$<br />
\xymatrix{<br />
& V_1 \ar[dd]^H\ar[rd]^{j_1|_{V_1}} \\<br />
B\ar[ru]^{i_1}\ar[rd]_{i_2} & & B \\<br />
& V_2 \ar[ru]_{j_2|_{V_2}}<br />
}<br />
$$<br />
<br />
{{endthm|Definition}}<br />
<br />
{{beginthm|Theorem|\cite{Kister1964|Theorem 2} }}<br />
Let $(E, B, i, j)$ be an $n$-dimensional microbundle. Then there is a neighbourhood of $i(B)$, $E_1 \subset E$ such that:<br />
# $E_1$ is the total space of a topological $\Rr^n$-bundle over $B$.<br />
# The inclusion $E_1 \to E$ is a microbundle isomorphism<br />
# If $E_2 \subset E$ is any other such neighbourhood of $i(B)$ then there is a $\Rr^n$-bundle isomorphism $(E_1 \to B) \cong (E_2 \to B)$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Talk:Microbundles_(Ex)Talk:Microbundles (Ex)2012-05-29T18:42:50Z<p>Marek Kaluba: </p>
<hr />
<div><wikitex>;<br />
Let us begin with the definition of [[Microbundle|microbundle]].<br />
{{beginthm|Definition|}}<br />
An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ such that there is a sequence $$B\xrightarrow{i} E\xrightarrow{j} B$$ and the following conditions hold.<br />
#$j\circ i=\id_B$<br />
#for all $x\in B$ there exist open neigbourhood $U\subset B$ and an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism $$h\colon V\to U\times \mathbb{R}^n.$$<br />
<br />
Moreover, the homeomorphism above must make the following diagram commute:<br />
$$<br />
\xymatrix{<br />
U \ar[d]^{i}\ar[r]& U\times\mathbb{R}^n \ar[d]^{p_1}\\<br />
V\ar[r]^{j} \ar[ur]^{h} & U,} <br />
$$<br />
where $p_1$ is projection on the first factor and $U$ is included as a $0$-section in $U\times \mathbb{R}^n$.<br />
{{endthm}}<br />
<br />
{{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1}}}}<br />
Let $M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle.<br />
{{endthm}}<br />
{{beginproof}} <br />
<br />
Let $M$ be a topological manifold. Then the composition $p_1\circ\Delta_M$ sends $x\mapsto (x,x)\mapsto x$, so the first condition in the definition is satisfied.<br />
<br />
To prove that the second condition is satisfied we need to use local chart around $x$.<br />
Choose $U$ to be one of the open sets coming from atlas of $M$ and let $\phi\colon U\to \mathbb{R}^n$ be associated chart. The obvious choice for neighbourhood $V\subset M\times M$ is to take $U\times U$. The first naive candidate for $h\colon V=U\times U\to U\times\mathbb{R}^n$ would be map $\id\times \phi$. However such $h$ fails to make the following diagram commute<br />
$$<br />
\xymatrix{<br />
U \ar[d]^{\Delta_M}\ar[r]^{\id\times \{0\}}& U\times\mathbb{R}^n \ar[d]^{p_1}\\<br />
V\ar[r]^{p_1} \ar[ur]^{h} & U,} <br />
$$<br />
since $(u,u)$ is mapped to $(u,\phi(u))$ and $\phi(u)$ doesn't necessarily be $0$ (well, it depends on our definition of an atlas of a manifold?). We need just a small adjustment: $h(u,v)=(u,h(u)-h(v))$.<br />
{{endproof}}<br />
<br />
<br />
{{beginthm|Exercise|{{citeD|Milnor1964|Theorem 2.2}}}}<br />
Let $M$ be a (paracompact!) smooth manifold. Show that $TM$ and $\xi_M$ are isomorphic microbundles.<br />
{{endthm}}<br />
{{beginproof}}<br />
We have two concurring definitions of micro-tangent bundle. The first is given by the exercise above, the second by forgeting about the vector bundle structure on $TM$ and treating it just as a microbundle $(TM, M, \pi,s_0)$ where $M\xrightarrow{s_0} TM$ is the zero section.<br />
<br />
However to show that these two definition agree we need a notion of microbundle isomorphism.<br />
<br />
{{beginthm|Definition}}<br />
Two microbundles $(E_n,X,i_n,j_n)$, $n=1,2$ over the same space $X$ are isomorphic if there exist neighbourhoods $V_1\subset E_1$ of $i_1(B)$ and $V_2\subset E_2$ of $i_2(B)$ and a homeomorphism $H\colon V_1\to V_2$ making the following diagram commute. <br />
$$<br />
\xymatrix{<br />
U \ar[d]^{i_1}\ar[r]^{i_2}& V_2 \ar[d]^{p_2}\\<br />
V_1\ar[r]^{p_1} \ar[ur]^{H} & U,} <br />
$$<br />
{{endthm|Definition}}<br />
<br />
In our case we have <br />
$$<br />
\xymatrix{<br />
U \ar[d]^{\Delta_M}\ar[r]^{s_0}& V \ar[d]^{p_2}\\<br />
U\times U\ar[r]^{p_1} \ar[ur]^{H} & U,} <br />
$$<br />
where $V\subset TM$ is an open neighbourhood of the zero section. Because of the vector bundle structure we may identify $V\cong U\times \mathbb{R}^n$ via local trivialisation.<br />
{{endproof}}<br />
<br />
</wikitex></div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Talk:Microbundles_(Ex)Talk:Microbundles (Ex)2012-05-29T18:25:38Z<p>Marek Kaluba: WiP</p>
<hr />
<div><wikitex>;<br />
Let us begin with the definition of [[Microbundle|microbundle]].<br />
{{beginthm|Definition|}}<br />
An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ such that there is a sequence $$B\xrightarrow{i} E\xrightarrow{j} B$$ and the following conditions hold.<br />
#$j\circ i=\id_B$<br />
#for all $x\in B$ there exist open neigbourhood $U\subset B$ and an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism $$h\colon V\to U\times \mathbb{R}^n.$$<br />
<br />
Moreover, the homeomorphism above must make the following diagram commute:<br />
$$<br />
\xymatrix{<br />
U \ar[d]^{i}\ar[r]& U\times\mathbb{R}^n \ar[d]^{p_1}\\<br />
V\ar[r]^{j} \ar[ur]^{h} & U,} <br />
$$<br />
where $p_1$ is projection on the first factor and $U$ is included as a $0$-section in $U\times \mathbb{R}^n$.<br />
{{endthm}}<br />
<br />
{{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1}}}}<br />
Let $M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle.<br />
{{endthm}}<br />
{{beginproof}} <br />
<br />
Let $M$ be a topological manifold. Then the composition $p_1\circ\Delta_M$ sends $x\mapsto (x,x)\mapsto x$, so the first condition in the definition is satisfied.<br />
<br />
To prove that the second condition is satisfied we need to use local chart around $x$.<br />
Choose $U$ to be one of the open sets coming from atlas of $M$ and let $\phi\colon U\to \mathbb{R}^n$ be associated chart. The obvious candidate for $V\subset M\times M$ is to take $U\times U$. The first naive candidate for $h\colon V=U\times U\to U\times\mathbb{R}^n$ would be map $\id\times \phi$. However such $h$ fails to make the following diagram commute<br />
$$<br />
\xymatrix{<br />
U \ar[d]^{\Delta_M}\ar[r]^{\id\times \{0\}}& U\times\mathbb{R}^n \ar[d]^{p_1}\\<br />
V\ar[r]^{p_1} \ar[ur]^{h} & U,} <br />
$$<br />
since $(u,u)$ is mapped to $(u,\phi(u))$ and $\phi(u)$ doesn't necessarily be $0$ (well, it depends on our definition of an atlas of a manifold?). We need just a small adjustment: $h(u,v)=(u,h(u)-h(v))$.<br />
{{endproof}}<br />
<br />
<br />
{{beginthm|Exercise|{{citeD|Milnor1964|Theorem 2.2}}}}<br />
Let $M$ be a (paracompact!) smooth manifold. Show that $TM$ and $\xi_M$ are isomorphic microbundles.<br />
{{endthm}}<br />
{{beginproof}}<br />
We have two concurring definitions of micro-tangent bundle. The first is given by the exercise above, the second by forgeting about the vector bundle structure on $TM$ and treating it just as a microbundle $(TM, M, \pi,s_0)$ where $M\xrightarrow{s_0} TM$ is the zero section.<br />
<br />
However to show that these two definition agree we need a notion of microbundle isomorphism.<br />
<br />
{{beginthm|Definition}}<br />
Two microbundles $(E_n,X,i_n,j_n)$, $n=1,2$ over the same space $X$ are isomorphic if there exist neighbourhoods $V_1\subset E_1$ of $i_1(B)$ and $V_2\subset E_2$ of $i_2(B)$ and a homeomorphism $H\colon V_1\to V_2$ making the following diagram commute. <br />
$$<br />
\xymatrix{<br />
U \ar[d]^{i_1}\ar[r]^{i_2}& V_2 \ar[d]^{p_2}\\<br />
V_1\ar[r]^{p_1} \ar[ur]^{H} & U,} <br />
$$<br />
{{endthm|Definition}}<br />
<br />
In our case<br />
<br />
<br />
{{endproof}}<br />
<br />
</wikitex></div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Talk:Microbundles_(Ex)Talk:Microbundles (Ex)2012-05-29T17:25:36Z<p>Marek Kaluba: </p>
<hr />
<div><wikitex>;<br />
Let us begin with the definition of microbundle.<br />
{{beginthm|Definition|}}<br />
An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ such that there is a sequence $$B\xrightarrow{i} E\xrightarrow{j} B$$ and the following conditions hold.<br />
#$j\circ i=\id_B$<br />
#for all $x\in B$ there exist open neigbourhood $U\subset B$ and an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism $$h\colon V\to U\times \mathbb{R}^n.$$<br />
<br />
Moreover, the homeomorphism above must make the following diagram commute:<br />
{{endthm}}<br />
<br />
{{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1, Theorem 2.2}}}}<br />
Let $M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle.<br />
{{endthm}}<br />
<br />
Let $M$ be a topological manifold. Then the composition $p_1\circ\Delta_M$ sends $x\mapsto (x,x)\mapsto x$, so the first condition in the definition is satisfied.<br />
<br />
To prove that the second condition is satisfied we need to use local chart around $x$.<br />
Choose $U$ to be one of the open sets coming from atlas of $M$ and let $\phi\colon U\to \mathbb{R}^n$ be associated chart. The obvious candidate for $V\subset M\times M$ is to take $U\times U$. Now the first naive candidate for $h\colon V=U\timesU\to U\times\mathbb{R}^n$ would be map $\id\times \phi$. However <br />
<br />
{{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1, Theorem 2.2}}}}<br />
Let $M$ be a smooth manifold. Show that $TM$ and $\xi_M$ are isomorphic microbundles.<br />
{{endthm}}<br />
</wikitex></div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Talk:Microbundles_(Ex)Talk:Microbundles (Ex)2012-05-29T17:11:50Z<p>Marek Kaluba: Created page with "<wikitex> Let us begin with the definition of microbundle. {{beginthm|Definition|}} An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ such that there is a sequence $$B..."</p>
<hr />
<div><wikitex><br />
Let us begin with the definition of microbundle.<br />
{{beginthm|Definition|}}<br />
An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ such that there is a sequence $$B\xrightarrow{i} E\xrightarrow{j} B$$ and the following conditions hold.<br />
*$j\circ i=\id_B$<br />
*for all $x\in B$ there exist open neigbourhood $U\subset B$ and an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism $$h\colon V\to U\times \mathbb{R}^n.$$<br />
{{endthm}}<br />
<br />
<br />
<br />
</wikitex></div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Fake_projective_spaces_in_dim_6_(Ex)Fake projective spaces in dim 6 (Ex)2012-04-02T22:15:26Z<p>Marek Kaluba: </p>
<hr />
<div><wikitex>;<br />
Let $M_B^8$ be a closed [[Milnor manifold]]. There exist a natural degree 1 normal map $M_B^8\to S^8$ which induces a degree 1 normal map $$f\colon\mathbb{C}P^4\#_mM_B^8\to \mathbb{C}P^4\#_m S^8\cong \mathbb{C}P^4.$$<br />
<br />
We may pull back the Hopf bundle $H_4\to \mathbb{C}P^4$ to $\mathbb{C}P^4\#_mM_B^8$. The sphere bundle of the Hopf bundle over sphere is the true sphere, hence we get a map:<br />
$$f|_\partial\colon S(f^*(H_4))\to S(H_4)=S^9.$$<br />
'''Prove the following:'''<br />
\begin{lemma}<br />
The map induced above is a degree 1 normal map.<br />
\end{lemma}<br />
<br />
Now, since we are working between odd dimensional manifolds we can by surgery below middle dimension assume that $f|_\partial$ bordant to a homotopy equivalence $g$. Thus by Poincare Conjecture it is indeed normally bordant to a PL-homeomorphism. Choose $W'\to S^9\times[0,1]$ to be such bordism.<br />
<br />
\begin{lemma} We can perform surgery on $$\bar{g}\colon W'\cup_\partial D(f^*(H_4))\to D(H_4)\cup_\partial S^9\times [0,1]\cong D(H_4)$$ to make it a homotopy equivalence. <br />
\end{lemma}<br />
<br />
'''Describe these surgeries.'''<br />
<br />
We obtain a manifold with boundary $(W^10,\partial W)$ homotopy homotopy equivalent to $D(H_4)$ with boundary PL-homeomorphic to $S^9$. Thus we may cone off common boundaries extending the homotopy equivalence at the same time. We define <br />
$$\widetilde{\mathbb{C}P^5}=W^{10}\cup_{S^9}D^{10}.$$<br />
<br />
\begin{lemma}<br />
$\widetilde{\mathbb{C}P^5}$ and $\mathbb{C}P^5$ are not homeomorphic.<br />
\end{lemma}<br />
<br />
'''Prove the above lemma.'''<br />
<br />
However, the construction above gives us a degree 1 normal map $$h\colon\widetilde{\mathbb{C}P^5}\to \mathbb{C}P^5.$$ We may pull back the canonical line bundle $H_5\to \mathbb{C}P^5$ over $\widetilde{\mathbb{C}P^5}$. By the Poincaré conjecture, the sphere bundle $S(h^*(H_5))=\partial D(h^*(H_5))$ is $PL$-homeomorphic to the sphere $S^{11}$. $\widetilde{\mathbb{C}P^6}=D(h^*(H_5))\cup_{S^{11}}D^{12}$ is a homotopy complex projective space obtained by coning off the boundary of the disk bundle.<br />
<br />
Similarly we may form the connected sum $\widetilde{\mathbb{C}P^5}\# M^{10}_A$, where $M^{10}_A$ is the [[Kervaire_sphere|Kervaire manifold]], and once again, we take pullback of $H_5$ line bundle via the natural degree 1 normal map. Now to obtain a homotopy equivalence $\partial D(f^*(H_5))\to \partial D(H_5)$ we have to use a little bit of surgery. <br />
<br />
'''Describe these surgeries.'''<br />
<br />
Nevertheless there exists a manifold $W^{12}$ such that $$D(f^*(H_5))\cup W^{12}\to D(H_5)$$ is a homotopy equivalence of manifolds with boundary. Again since the boundary is already $PL$-homeomorphic to a sphere we may extend the homotopy equivalence after coning off the boundaries.<br />
<br />
{{beginthm|Lemma}}<br />
$\mathbb{C}P^6$, $\widetilde{\mathbb{C}P^6}$ and $f^*(H_5)\cup W^{12}\cup_{S^{11}}D^{12}$, are topologically distinct homotopy projective spaces.<br />
{{endthm}}<br />
<br />
The aim of this exercise is to '''write full details of proof of the above Lemma''' (which is Lemma 8.24 form \cite{Madsen&Milgram1979}).<br />
<br />
A sketch of this proof can be found in the book, on page 170.<br />
</wikitex><br />
== References ==<br />
{{#RefList:}}<br />
[[Category:Exercises]]<br />
[[Category:Exercises without solution]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Talk:Fake_projective_spaces_in_dim_6_(Ex)Talk:Fake projective spaces in dim 6 (Ex)2012-04-02T22:13:36Z<p>Marek Kaluba: Created page with "<wikitex>; Any idea for different symbol for fake complex projective space? \widetilda looks too ugly... </wikitex>"</p>
<hr />
<div><wikitex>;<br />
Any idea for different symbol for fake complex projective space?<br />
\widetilda looks too ugly...<br />
<br />
</wikitex></div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Fake_projective_spaces_in_dim_6_(Ex)Fake projective spaces in dim 6 (Ex)2012-04-02T22:12:07Z<p>Marek Kaluba: Added beginning of the construction of hCP^5. Expanded exercise range and made it more modular.</p>
<hr />
<div><wikitex>;<br />
Let $M_B^8$ be a closed [[Milnor manifold]]. There exist a natural degree 1 normal map $M_B^8\to S^8$ which induces a degree 1 normal map $$f\colon\mathbb{C}P^4\#_mM_B^8\to \mathbb{C}P^4\#_m S^8\cong \mathbb{C}P^4.$$<br />
<br />
We may pull back the Hopf bundle $H_4\to \mathbb{C}P^4$ to $\mathbb{C}P^4\#_mM_B^8$. The sphere bundle of the Hopf bundle over sphere is the true sphere, hence we get a map:<br />
$$f|_\partial\colon S(f^*(H_4))\to S(H_4)=S^9.$$<br />
'''Prove the following:'''<br />
\begin{lemma}<br />
The map induced above is a degree 1 normal map.<br />
\end{lemma}<br />
<br />
Now, since we are working between odd dimensional manifolds we can by surgery below middle dimension assume that $f|_\partial$ bordant to a homotopy equivalence $g$. Thus by Poincare Conjecture it is indeed normally bordant to a PL-homeomorphism. Choose $W'\to S^9\times[0,1]$ to be such bordism.<br />
<br />
\begin{lemma} We can perform surgery on $$\bar{g}\colon W'\cup_\partial D(f^*(H_4))\to D(H_4)\cup_\partial S^9\times [0,1]\cong D(H_4)$$ to make it a homotopy equivalence. <br />
\end{lemma}<br />
<br />
'''Describe these surgeries.'''<br />
<br />
We obtain a manifold with boundary $(W^10,\partial W)$ homotopy homotopy equivalent to $D(H_4)$ with boundary PL-homeomorphic to $S^9$. Thus we may cone off common boundaries extending the homotopy equivalence at the same time. We define <br />
$$\widetilde{\mathbb{C}P^5}=W^{10}\cup_{S^9}D^{10}.$$<br />
<br />
\begin{lemma}<br />
$\widetilde{\mathbb{C}P^5}$ and $\mathbb{C}P^5$ are not homeomorphic.<br />
\end{lemma}<br />
<br />
'''Prove the above lemma.'''<br />
<br />
The above construction gives us a degree 1 normal map $$h\colon\widetilde{\mathbb{C}P^5}\to \mathbb{C}P^5.$$ We may pull back the canonical line bundle $H_5\to \mathbb{C}P^5$ over $\widetilde{\mathbb{C}P^5}$. By the Poincaré conjecture, the sphere bundle $S(h^*(H_5))=\partial D(h^*(H_5))$ is $PL$-homeomorphic to the sphere $S^{11}$. $\widetilde{\mathbb{C}P^6}=D(h^*(H_5))\cup_{S^{11}}D^{12}$ is a homotopy complex projective space obtained by coning off the boundary of the disk bundle.<br />
<br />
Similarly we may form the connected sum $\widetilde{\mathbb{C}P^5}\# M^{10}_A$, where $M^{10}_A$ is the [[Kervaire_sphere|Kervaire manifold]], and once again, we take pullback of $H_5$ line bundle via the natural degree 1 normal map. Now to obtain a homotopy equivalence $\partial D(f^*(H_5))\to \partial D(H_5)$ we have to use a little bit of surgery. <br />
<br />
'''Describe these surgeries.'''<br />
<br />
Nevertheless there exists a manifold $W^{12}$ such that $$D(f^*(H_5))\cup W^{12}\to D(H_5)$$ is a homotopy equivalence of manifolds with boundary. Again since the boundary is already $PL$-homeomorphic to a sphere we may extend the homotopy equivalence after coning off the boundaries.<br />
<br />
{{beginthm|Lemma}}<br />
$\mathbb{C}P^6$, $\widetilde{\mathbb{C}P^6}$ and $f^*(H_5)\cup W^{12}\cup_{S^{11}}D^{12}$, are topologically distinct homotopy projective spaces.<br />
{{endthm}}<br />
<br />
The aim of this exercise is to '''write full details of proof of the above Lemma''' (which is Lemma 8.24 form \cite{Madsen&Milgram1979}).<br />
<br />
A sketch of this proof can be found in the book, on page 170.<br />
</wikitex><br />
== References ==<br />
{{#RefList:}}<br />
[[Category:Exercises]]<br />
[[Category:Exercises without solution]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Fake_projective_spaces_in_dim_6_(Ex)Fake projective spaces in dim 6 (Ex)2012-03-23T12:59:00Z<p>Marek Kaluba: </p>
<hr />
<div><wikitex>;<br />
<br />
There is a degree 1 normal map $$h\colon\widetilde{\mathbb{C}P^5}\to \mathbb{C}P^5.$$ We may pull back the canonical line bundle $H_5\to \mathbb{C}P^5$ over $\widetilde{\mathbb{C}P^5}$. By the Poincaré conjecture, the sphere bundle $S(h^*(H_5))=\partial D(h^*(H_5))$ is $PL$-homeomorphic to the sphere $S^{11}$. $\widetilde{\mathbb{C}P^6}=D(h^*(H_5))\cup_{S^{11}}D^{12}$ is a homotopy complex projective space obtained by coning off the boundary of the disk bundle.<br />
<br />
Similarly we may form the connected sum $\widetilde{\mathbb{C}P^5}\# M^{10}_A$, where $M^{10}_A$ is the [[Kervaire_sphere|Kervaire manifold]], and once again, we take pullback of $H_5$ line bundle via the natural degree 1 normal map. Now to obtain a homotopy equivalence $\partial D(f^*(H_5))\to \partial D(H_5)$ we have to use a little bit of surgery. Nevertheless there exists a manifold $W^{12}$ such that $$D(f^*(H_5))\cup W^{12}\to D(H_5)$$ is a homotopy equivalence of manifolds with boundary. Again since the boundary is already $PL$-homeomorphic to a sphere we may extend the homotopy equivalence after coning off the boundaries.<br />
<br />
{{beginthm|Lemma}}<br />
$\mathbb{C}P^6$, $\widetilde{\mathbb{C}P^6}$ and $f^*(H_5)\cup W^{12}\cup_{S^{11}}D^{12}$, are topologically distinct homotopy projective spaces.<br />
{{endthm}}<br />
<br />
The aim of this exercise is to write full details of proof of Lemma 8.24 form \cite{Madsen&Milgram1979}.<br />
<br />
A sketch of this proof can be found in the book, on page 170.<br />
</wikitex><br />
== References ==<br />
{{#RefList:}}<br />
[[Category:Exercises]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Regensburg_Surgery_Blockseminar_2012:_ExercisesRegensburg Surgery Blockseminar 2012: Exercises2012-03-23T12:40:23Z<p>Marek Kaluba: </p>
<hr />
<div>This page lists the exercises for consideration during the Blockseminar. Participants are encouraged submit exercises related to their own talks and to work on solutions to all exercises.<br />
<br />
* [[2012 SBR General information|General information]]<br />
* [[2012 SBR Program|Program]]<br />
<br />
* [A] Talks 1-3: [[Whitehead torsion (Ex)]], [[Whitehead torsion II (Ex)]], [[Whitehead torsion III (Ex)]], [[Whitehead torsion IV (Ex)]]<br />
* [B] Talks 4&5: [[Poincaré duality (Ex)]], [[Poincaré duality II (Ex)]], [[Poincaré duality III (Ex)]], [[Poincaré duality IV (Ex)]], [[Spivak normal fibration (Ex)]]<br />
*[C] Talks 6-8: [[Regular homotopy group of immersions (Ex)]], [[Normal maps and submanifolds (Ex)]], [[Sphere bundles and spin (Ex)]]<br />
*[D] Talks 9-11: [[Homology braid (Ex)]], [[Middle-dimensional surgery kernel (Ex)]], [[Even dimensional surgery obstruction (Ex)]], [[Quadratic Formations (Ex)]]<br />
[[Category:2012 Surgery Blockseminar Regensburg]]<br />
*[E] Talks 12-14:<br />
*[F] Talks 15-18: [[Almost framed bordism (Ex)]], [[Exotic spheres and chirality (Ex)]], [[Fake projective spaces in dim 6 (Ex)]], [[Splitting invariants (Ex)]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Fake_projective_spaces_in_dim_6Fake projective spaces in dim 62012-03-23T12:39:46Z<p>Marek Kaluba: moved Fake projective spaces in dim 6 to Fake projective spaces in dim 6 (Ex): Guidelines</p>
<hr />
<div>#REDIRECT [[Fake projective spaces in dim 6 (Ex)]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Fake_projective_spaces_in_dim_6_(Ex)Fake projective spaces in dim 6 (Ex)2012-03-23T12:39:46Z<p>Marek Kaluba: moved Fake projective spaces in dim 6 to Fake projective spaces in dim 6 (Ex): Guidelines</p>
<hr />
<div><wikitex>;<br />
<br />
There is a degree $1$ normal map $$h\colon\widetilde{\mathbb{C}P^5}\to \mathbb{C}P^5.$$ We may pull back the canonical line bundle $H_5\to \mathbb{C}P^5$ over $\widetilde{\mathbb{C}P^5}$. By the Poincar\'{e} conjecture, the sphere bundle $S(h^*(H_5))=\partial D(h^*(H_5))$ is $PL$-homeomorphic to the sphere $S^{11}$. $\widetilde{\mathbb{C}P^6}=D(h^*(H_5))\cup_{S^{11}}D^{12}$ is a homotopy complex projective space obtained by coning off the boundary of the disk bundle.<br />
<br />
Similarly we may form the connected sum $\widetilde{\mathbb{C}P^5}\# M^{10}_A$, where $M^{10}_A$ is the [[Kervaire_sphere|Kervaire manifold]], and once again, we take pullback of $H_5$ line bundle via the natural degree $1$ normal map. Now to obtain a homotopy equivalence $\partial D(f^*(H_5))\to \partial D(H_5)$ we have to use a little bit of surgery. Nevertheless there exists a manifold $W^{12}$ such that $$D(f^*(H_5))\cup W^{12}\to D(H_5)$$ is a homotopy equivalence of manifolds with boundary. Again since the boundary is already $PL$-homeomorphic to a sphere we may extend the homotopy equivalence after coning off the boundaries.<br />
<br />
{{beginthm|Lemma}}<br />
$\mathbb{C}P^6$, $\widetilde{\mathbb{C}P^6}$ and $f^*(H_5)\cup W^{12}\cup_{S^{11}}D^{12}$, are topologically distinct homotopy projective spaces.<br />
{{endthm}}<br />
<br />
The aim of this exercise is to write full details of proof of Lemma 8.24 form \cite{Madsen&Milgram1979}.<br />
<br />
A sketch of this proof can be found in the book, on page 170.<br />
<br />
</wikitex><br />
== References ==<br />
{{#RefList:}}<br />
[[Category:Exercises]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Regensburg_Surgery_Blockseminar_2012:_ExercisesRegensburg Surgery Blockseminar 2012: Exercises2012-03-23T12:27:12Z<p>Marek Kaluba: </p>
<hr />
<div>This page lists the exercises for consideration during the Blockseminar. Participants are encouraged submit exercises related to their own talks and to work on solutions to all exercises.<br />
<br />
* [[2012 SBR General information|General information]]<br />
* [[2012 SBR Program|Program]]<br />
<br />
* [A] Talks 1-3: [[Whitehead torsion (Ex)]], [[Whitehead torsion II (Ex)]], [[Whitehead torsion III (Ex)]], [[Whitehead torsion IV (Ex)]]<br />
* [B] Talks 4&5: [[Poincaré duality (Ex)]], [[Poincaré duality II (Ex)]], [[Poincaré duality III (Ex)]], [[Poincaré duality IV (Ex)]], [[Spivak normal fibration (Ex)]]<br />
*[C] Talks 6-8: [[Regular homotopy group of immersions (Ex)]], [[Normal maps and submanifolds (Ex)]], [[Sphere bundles and spin (Ex)]]<br />
*[D] Talks 9-11: [[Homology braid (Ex)]], [[Middle-dimensional surgery kernel (Ex)]], [[Even dimensional surgery obstruction (Ex)]], [[Quadratic Formations (Ex)]]<br />
[[Category:2012 Surgery Blockseminar Regensburg]]<br />
*[E] Talks 12-14:<br />
*[F] Talks 15-18: [[Almost framed bordism (Ex)]], [[Exotic spheres and chirality (Ex)]], [[Fake projective spaces in dim 6]], [[Splitting invariants (Ex)]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Splitting_invariants_(Ex)Splitting invariants (Ex)2012-03-23T12:22:22Z<p>Marek Kaluba: Created page with "<wikitex>; Prove that two maps $f_1,f_2 \colon \mathbb{C}P^n \to G/PL$ are homotopic iff their splitting invariants agree for $2 \leq i \leq n$. Use the exact sequence $$L_{2..."</p>
<hr />
<div><wikitex>;<br />
Prove that two maps $f_1,f_2 \colon \mathbb{C}P^n \to G/PL$ are<br />
homotopic iff their splitting invariants agree for $2 \leq i \leq n$.<br />
<br />
Use the exact sequence<br />
$$L_{2k}(\mathbb{Z}) \to [\mathbb{C}P^k,G/PL] \to<br />
[\mathbb{C}P^{k-1},G/PL] \to 0$$<br />
and the fact that the surgery obstruction map $$\theta \colon<br />
[\mathbb{C}P^k, G/PL] \to L_{2k}(\mathbb{Z})$$ splits the above sequence for $k>2$. Additionally $[\mathbb{C}P^2,G/PL]=\mathbb{Z}$ and the isomorphism is given by the surgery obstruction map.<br />
</wikitex><br />
== References ==<br />
{{#RefList:}}<br />
[[Category:Exercises]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Fake_projective_spaces_in_dim_6_(Ex)Fake projective spaces in dim 6 (Ex)2012-03-23T12:13:05Z<p>Marek Kaluba: Created page with "<wikitex>; There is a degree $1$ normal map $$h\colon\widetilde{\mathbb{C}P^5}\to \mathbb{C}P^5.$$ We may pull back the canonical line bundle $H_5\to \mathbb{C}P^5$ over $\wi..."</p>
<hr />
<div><wikitex>;<br />
<br />
There is a degree $1$ normal map $$h\colon\widetilde{\mathbb{C}P^5}\to \mathbb{C}P^5.$$ We may pull back the canonical line bundle $H_5\to \mathbb{C}P^5$ over $\widetilde{\mathbb{C}P^5}$. By the Poincar\'{e} conjecture, the sphere bundle $S(h^*(H_5))=\partial D(h^*(H_5))$ is $PL$-homeomorphic to the sphere $S^{11}$. $\widetilde{\mathbb{C}P^6}=D(h^*(H_5))\cup_{S^{11}}D^{12}$ is a homotopy complex projective space obtained by coning off the boundary of the disk bundle.<br />
<br />
Similarly we may form the connected sum $\widetilde{\mathbb{C}P^5}\# M^{10}_A$, where $M^{10}_A$ is the [[Kervaire_sphere|Kervaire manifold]], and once again, we take pullback of $H_5$ line bundle via the natural degree $1$ normal map. Now to obtain a homotopy equivalence $\partial D(f^*(H_5))\to \partial D(H_5)$ we have to use a little bit of surgery. Nevertheless there exists a manifold $W^{12}$ such that $$D(f^*(H_5))\cup W^{12}\to D(H_5)$$ is a homotopy equivalence of manifolds with boundary. Again since the boundary is already $PL$-homeomorphic to a sphere we may extend the homotopy equivalence after coning off the boundaries.<br />
<br />
{{beginthm|Lemma}}<br />
$\mathbb{C}P^6$, $\widetilde{\mathbb{C}P^6}$ and $f^*(H_5)\cup W^{12}\cup_{S^{11}}D^{12}$, are topologically distinct homotopy projective spaces.<br />
{{endthm}}<br />
<br />
The aim of this exercise is to write full details of proof of Lemma 8.24 form \cite{Madsen&Milgram1979}.<br />
<br />
A sketch of this proof can be found in the book, on page 170.<br />
<br />
</wikitex><br />
== References ==<br />
{{#RefList:}}<br />
[[Category:Exercises]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/3-manifolds3-manifolds2011-01-28T18:39:29Z<p>Marek Kaluba: /* Type III: infinite non-cyclic fundamental group */</p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
In the 3-dimensional setting there is no distinction between smooth, PL and topological manifolds neccesary; the categories of smooth, PL and topological manifolds are equivalent (TODO ref). A lot of techniques have been developed in the last century to study 3-manifolds but most of them are very special and don't generalise to higher dimensions. One key idea is to decompose manifolds along incompressible surfaces into smaller pieces, to which certain geometric models apply. <br />
A great progress was made in with the proof of the Poincaré conjecture and Thurton's geometrization conjecture by Perelman in 2003.<br />
[[Image:Phomsphere.jpg|thumb|150px| The universal cover of the famous [[Poincaré homology sphere]] is $S^3$ - here a view of the induced tesselation]]<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<br />
<wikitex>;<br />
Basic examples are $\mathbb{R}^3, S^3, S^1 \times S$ with $S$ any surface. <br />
Important types of 3-manifolds are [[Wikipedia:Haken_manifold|Haken-Manifolds]], [[Wikipedia:Seifert_fibre_spaces|Seifert-Manifolds]], [[Wikipedia:Lens_spaces|3-dimensional lens spaces]], [[Wikipedia:Torus_bundle|Torus-bundles and Torus semi-bundles]].<br />
<br />
There are two topological processes to join 3-manifolds to get a new one.<br />
The first is the connected sum of two manifolds $M_1$ and $M_2$. Choose embeddings $f_1:D^3\rightarrow M_1$ and $f_2:D^3\rightarrow M_2$, remove the interior of $f_1(D^3)$ and $f_2(D^3)$ and glue $M_1$ and $M_2$ together along the boundaries $f_1(S^3)$ and $f_2(S^3)$.<br />
The second uses incompressible surfaces. Let $M$ be manifold and $S\subset M$ a surface. $S$ is incompressible, if there is no disk $D$ in $M$ with $D\cap S=\partial D$.<br />
The torus sum is the process which glues incompressible tori boundary components together.<br />
<br />
(TODO What is incompressibility needed? / What is is good for/ What happen if one takes a compressible surface ?)<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
In the 3-dimensional world the fundamental group is a powerful invariant to distiguish manifolds.<br />
It determines already all homology groups: <br />
* $H_1(M)$ = abelization of $\pi_1(M)$. <br />
* $H_2(M) = H^1(M) = H_1(M)/$torsion<br />
* $H_3(M) = \Zz$<br />
* $H_n(M) = 0$ for $n > 3$ <br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
By reversing the process of connected and torus sum every 3-manifold can be decomposed into pieces which admit a geometric structure. We describe the details in the following.<br />
<br />
=== Prime decomposition === <br />
<wikitex>;<br />
{{beginthm|Definition}}<br />
A manifold $M$ is called prime, if it can't be written as a non-trivial connected sum, i.e. $M=M_1 \# M_2$ implies $M_1 = S^3$ or $M_2 = S^3$.<br />
A manifold $M$ is called irreducible if every embedded $S^2$ bounds a ball, i.e. the embedding extends to an embedding of $D^3$<br />
{{endthm}}<br />
Irreducibility is only slightly stronger than being prime. A orientable prime 3-manifold is either $S^2 \times S^1$ or every embedded 2-sphere bounds a ball.<br />
<br />
\begin{theorem}[Kneser]<br />
Every orientable, compact 3-manifold $M$ has a decomposition $M=P_1 \# \ldots \# P_n$ into prime manifolds $P_i$ unique up to ordering and $S^3$ summands.<br />
\end{theorem} <br />
<br />
A orientable prime 3-manifold is either $S^2 \times S^1$ or every embedded 2-sphere bounds a ball, in which case the manifold is called irreducible.<br />
<br />
Van Kampen's theorem tells you, that $\pi_1(M \# N)=\pi_1(M)*\pi_1(N)$. Hence any 3-manifold, whose fundamental group cannot be written as a free product of two nontrivial subgroups, can only be written as the connected sum of another 3-manifold with a simply connected 3-manifold. By the Poincaré conjecture a simply connected 3-manifold is already homeomorphic to $S^3$. Hence each such manifold is prime.<br />
<br />
Prime 3-manifolds can be distinguished by their fundamental groups into the following 3 types:<br />
</wikitex><br />
==== Type I: finite fundamental group==== <br />
<wikitex>;<br />
The universal cover $\tilde{M}$ is a simply-connected 3-manifold. As the fundamental group already determines the homology of a oriented, closed compact 3-manifold, it has to be a homology sphere. Using the Hurewicz-theorem, its fundamental class is represented by a degree 1 map $S^3 \rightarrow \tilde{M}$. This map induces isomorphisms on the homology and on the fundamental group. Hence it is a weak homotopy equivalence, and hence a homotopy equivalence by Whitehead's theorem (ref?).<br />
Hence every prime $3$-manifold with finite fundamental group arises as the quotient of a homotopy sphere by a free action of a finite group.<br />
With the use of the Poincaré conjecture every homotopy 3-sphere is homeomorphic to $S^3$ and we can write $M=S^3/\Gamma$. If $\Gamma$ is cyclic $M$ is known as lens space (ref). <br />
</wikitex><br />
==== Type II: infinite cyclic fundamental group ====<br />
<wikitex>;<br />
$S^1\times S^2$ is the only orientable closed prime 3-manifold of this type. Futhermore it is the only not irreducible prime manifold. (TODO: proof/ref)<br />
</wikitex><br />
<br />
==== Type III: infinite non-cyclic fundamental group ====<br />
<wikitex>;<br />
Such a manifold $M$ is always aspherical (TODO ref). The sphere theorem states, that every map $S^2\rightarrow M$ is homotopic to an embedding; and - as $M$ is irreducible - it is nullhomotopic. Hence $\pi_2(M)=0$. Consider the universal covering $\tilde{M}$ of $M$. Its first homology vanishes as it is simply connected. The long exact sequence of homotopy groups of the fibration $\pi_1(M)\rightarrow \tilde{M}\rightarrow M$ gives a isomorphism $\pi_2(M)\cong \pi_2(\tilde{M})$. Hence by Hurewicz' theorem $H_2(\tilde{M})=0$. Furthermore $H_3(\tilde{M})=0$, as $M$ is noncompact. Applying Hurewicz theorem again we get that all homotopy groups of $\tilde{M}$ vanish and hence by Whitehead's theorem $\tilde{M}$ is contractible. This means that $M$ is apherical. <br />
Hence the homotopy type of a prime 3-manifold with infinite non-cyclic fundamental group is uniquely determined by its fundamental group.<br />
Furthermore not every group can occur as a fundamental group of a prime 3-manifold. The equivariant cellular chain complex of $\tilde{M}$ is a projective resolution of the trivial $\Zz[\pi_1(M)]$-module $\Zz$. Hence ....<br />
For any subgroup $F\le \pi_1(M)$ the space $\tilde{M}/F$ is a finite-dimensional model for $K(F,1)$. For example a finite group cannot have such a model (by group homology ref) and hence $\pi_1(M)$ must be torsionfree. Furthermore it is a Poincaré duality group (link).<br />
</wikitex><br />
<br />
=== Torus decomposition ===<br />
<wikitex>;<br />
According to the previous section it remains to classify irreducible prime 3-manifolds. <br />
After cutting along spheres which don't bound balls as far as possible the next canonical step is to consider incompressible tori which are disjoint from the boundary. <br />
<br />
\begin{theorem}[Jacob-Shalen, Johannson]<br />
If $M$ is an irreducible compact orientable manifold, then there is a collection of disjoint incompressible tori $T_1, \ldots ,T_n$ in $M$ such that splitting $M$ along the union of these tori produces manifolds $M_i$ which are either [[Wikipedia:Seifert_fiber_space|Seifert-fibered]] or atoroidal, i.e. every incompressible torus in $M_i$ is isotopic to a torus component of $\partial M_i$. Furthermore, a minimal such collection of tori $T_j$ is unique up to isotopy in $M$.<br />
\end{theorem}<br />
<br />
Thurston's geometrization conjectures states that all the pieces we get by this JSJ-decomposition admit one of eight possible geometric structures:<br />
There is a list of eight simply connected Riemannian manifolds - the so called model geometries. A geometric structure on $M$ is the choice of a Riemannian metric on $M$, with the property that its universal covering $ \tilde{M}$ equipped with the pull-back metric is isometric to one of the eight model geometries. It might a priori be easier to classify all cocompact actions on the several model geometries.<br />
<br />
The Seifert-fibered pieces are well understood since the work of Seifert in the 30s (TODO: mention classification theorem). The atoroidal pieces are described by the following Hyperbolization theorem which was stated by Thurston (ref) and proven by Perelman.<br />
\begin{thm}<br />
Every irreducible atoroidal closed 3-manifold that is not Seifert-fibred is hyperbolic.<br />
\end{thm}<br />
</wikitex><br />
<br />
=== Dehn surgery ===<br />
<wikitex><br />
Dehn surgery is a way of constructing (TODO oriented ? neccesary) 3-manifolds. Given a [[Wikipedia:Link_%28knot_theory%29 | link]] in a $3$-manifold $N$<br />
$$L: \coprod_{i=1}^n S^1\rightarrow N,$$<br />
and a choice of a tubular neighborhood of $L$ <br />
$$L': \coprod_{i=1}^n S^1\times D^2\rightarrow N\mbox{ with }L'(x,0)=L(x)$$.<br />
(This choice essentially is the choice of a trivialization of the normal bundle; TODO find a correct formulation for this).<br />
This gives us a family of embedded, disjoint, full tori. The idea of Dehn surgery is to remove these Tori and glue them back in using a twist.\\<br />
Let us restrict to the case with only one solid torus $L':S^1\times D^2\rightarrow N$. <br />
Choose any self-homeomorphism $f$ of the torus $S^1\times S^1$. The result of the Dehn surgery at $L$ with the twist $f$ is defined as <br />
$$N_{f,L'}:=N\setminus L'(S^1\times \mathring{D}^2) \cup_f S^1\times D^2=N\setminus L'(S^1\times \mathring{D}^2) \amalg S^1\times D^2/\sim,$$<br />
where the equivalence relation identifies for $(x,y)\in S^1\times S^1$ the points $L'(x,y)$ in the left component and $f(x,y)$ in the right component.<br />
If $f$ is the coordinate flipping, Dehn surgery is nothing but usual codimension $2$ surgery.<br />
<br />
\begin{lemma}<br />
Suppose $f,f'\in \Homeo(T^2)$ are isotopic and let $L':S^1\times D^2 \rightarrow N$ be any embedding of the full Torus in a $3$-Manifold $N$.Then $N_{f,L'}$ and $N_{f',L'}$ are homeomorphic.<br />
\end{lemma}<br />
\begin{proof}<br />
Let $j:T^2\times [0;1] \rightarrow T^2$ be an isotopy from $f$ to $f'$. This gives a homeomorphism:<br />
$$ J:T^2\times [0;1]\rightarrow T^2\times [0;1] \qquad (x,y,t)\mapsto (j(x,y,t),t). $$<br />
(TODO: is its inverse $(x,y,t)\mapsto (j(-,-,t)^{-1}(x,y),t)$ continuous ?).<br />
The idea is to grab some additional space, where one can use the map $J$. TODO<br />
\end{proof}<br />
TODO formulate a lemma, that M_{f,L'} also only depends on the isotopy class of $L'$ (which is hopefully true).<br />
Hence, we have to classify all self-homeomorphisms of $T^2$ up to isotopy.<br />
\begin{lemma}<br />
Every self-homeomorphism of $T^2$ is isotopic to exactly one homeomorphism of the shape <br />
$$f_A:\Rr^2/\Zz^2 \rightarrow \Rr^2 /\Zz^2 \qquad \left(\begin{array}{c}x\\y\end{array}\right)\mapsto A\cdot\left(\begin{array}{c}x\\y\end{array}\right),$$<br />
where $A\in GL_2(\Zz)$ (reference of proof). <br />
\end{lemma}<br />
\begin{proof} Since the torus is a $K\left(\Zz^2,1\right)$-space, we have that $\pi_0 Map\left(T^2,T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$ is an isomorphism. Homotopic surface homeomorphisms are isotopic (Reference?). Thus the restriction $\pi_0Homeo\left(T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$ is injective. Moreover, each $A\in Hom\left(\Zz^2,\Zz^2\right)$ is realised by $f_A$, therefore $\pi_0Homeo\left(T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$ is also surjective. <br />
TODO find a reference in ANY source about the mapping class group<br />
\end{proof}<br />
The next lemma tells us, that composition of self-homeomorphisms corresponds to two successive Dehn surgeries.<br />
\begin{lemma}<br />
Let $f,g\in \Homeo(T^2)$ be given and let $L':S^1\times D^2\rightarrow N$ is an embedding of the full torus in a 3-manifold. Then we have map <br />
$$L'':S^1 \times D^2 \rightarrow N\setminus L'(S^1\times \mathring{D}^2) \cup_f S^1\times D^2=N_{f,L'}$$<br />
given by the map $S^1\times D^2\rightarrow S^1 \times D^2 \quad (x,y)(x,y/2)$ postcomposed with the canonical inclusion in the second coordinate.<br />
Then $(N_{f,L'})_{g,L''} \cong N_{f\circ g,L'}$. TODO right order of composition ? We will see in the proof.<br />
\end{lemma}<br />
\beg{proof}<br />
\end{proof}<br />
<br />
We have to find out, which self-homeomorphisms of the torus don't change the homeomorphism type of the manifold.<br />
\begin{lemma} Consider a matrix of the form $\left( \begin{array}{cc} 1 & 0 \\ k & 1 \end{array}\right)$ and let $L':S^1\times D^2 \rightarrow N$ be any eembedding. Then $N_{f_A,L'}\cong N$.<br />
\end{lemma}<br />
TODO are there any orientation reversing homeos, that also extend ? Think so. Also add them here.<br />
\begin{proof}<br />
The homeomorphism $f_A \in \Homeo(T^2)$ extends to a homeomorphism of $\bar{f_A}\in \Homeo(S^1\times D^2)$:<br />
$$\bar{f_A}(x,y):=(x,x^ky), $$<br />
where $x\in S^1 = \{z\in \Cc| |z|=1\}, y\in D^2=\{y\in\Cc||y|\le 1\}$.<br />
Using this homeomorphism one can define a homeomorphism from $N_f$ to $N$:<br />
$$N = N\setminus L(S^1\times \mathring{D}^2)\cup_1 S^1\times D^2 \rightarrow N_{f_A}=N\setminus L(S^1 \times \mathring{D}^2)\cup_{f_A} S^1\times D^2$$<br />
given by the identity on the left component and $\bar{f_A}$ on the right component.<br />
\end{proof}<br />
<br />
Together with (link to comment about composition), this tells us, that $N_{f_A}$ really only depends on the coset $A\cdot \left(\begin{array}{cc}1&*\\0&1\end{array}\right)$ (TODO check right or left coset). This coset is uniquely determined by the image $(p,q)$ of $(1,0)$ with $p$ and $q$ coprime.<br />
<br />
The ratio $p/q$ is called the surgery coefficient. (TODO what is the quotient good for ?)<++><br />
<br />
<br />
TODO does the result give different manifolds.<br />
<br />
TODO does the result only depend on the isotopy class of the link. <br />
<br />
Every compact (oriented /able, neccesary ?) 3-manifold might be obtained from $S^3$ by a Dehn surgery along a link (TODO ref). Of course this does not satisfy to classify 3-manifolds without having a good classification of links in $S^3$. <br />
<br />
<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
\cite{Scott1983}, \cite{Thurston1997}, \cite{Hatcher2000}, \cite{Hempel1976}<br />
{{#RefList:}}<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/3-manifolds3-manifolds2011-01-28T18:38:47Z<p>Marek Kaluba: /* Type III: infinite non-cyclic fundamental group */</p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
In the 3-dimensional setting there is no distinction between smooth, PL and topological manifolds neccesary; the categories of smooth, PL and topological manifolds are equivalent (TODO ref). A lot of techniques have been developed in the last century to study 3-manifolds but most of them are very special and don't generalise to higher dimensions. One key idea is to decompose manifolds along incompressible surfaces into smaller pieces, to which certain geometric models apply. <br />
A great progress was made in with the proof of the Poincaré conjecture and Thurton's geometrization conjecture by Perelman in 2003.<br />
[[Image:Phomsphere.jpg|thumb|150px| The universal cover of the famous [[Poincaré homology sphere]] is $S^3$ - here a view of the induced tesselation]]<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<br />
<wikitex>;<br />
Basic examples are $\mathbb{R}^3, S^3, S^1 \times S$ with $S$ any surface. <br />
Important types of 3-manifolds are [[Wikipedia:Haken_manifold|Haken-Manifolds]], [[Wikipedia:Seifert_fibre_spaces|Seifert-Manifolds]], [[Wikipedia:Lens_spaces|3-dimensional lens spaces]], [[Wikipedia:Torus_bundle|Torus-bundles and Torus semi-bundles]].<br />
<br />
There are two topological processes to join 3-manifolds to get a new one.<br />
The first is the connected sum of two manifolds $M_1$ and $M_2$. Choose embeddings $f_1:D^3\rightarrow M_1$ and $f_2:D^3\rightarrow M_2$, remove the interior of $f_1(D^3)$ and $f_2(D^3)$ and glue $M_1$ and $M_2$ together along the boundaries $f_1(S^3)$ and $f_2(S^3)$.<br />
The second uses incompressible surfaces. Let $M$ be manifold and $S\subset M$ a surface. $S$ is incompressible, if there is no disk $D$ in $M$ with $D\cap S=\partial D$.<br />
The torus sum is the process which glues incompressible tori boundary components together.<br />
<br />
(TODO What is incompressibility needed? / What is is good for/ What happen if one takes a compressible surface ?)<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
In the 3-dimensional world the fundamental group is a powerful invariant to distiguish manifolds.<br />
It determines already all homology groups: <br />
* $H_1(M)$ = abelization of $\pi_1(M)$. <br />
* $H_2(M) = H^1(M) = H_1(M)/$torsion<br />
* $H_3(M) = \Zz$<br />
* $H_n(M) = 0$ for $n > 3$ <br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
By reversing the process of connected and torus sum every 3-manifold can be decomposed into pieces which admit a geometric structure. We describe the details in the following.<br />
<br />
=== Prime decomposition === <br />
<wikitex>;<br />
{{beginthm|Definition}}<br />
A manifold $M$ is called prime, if it can't be written as a non-trivial connected sum, i.e. $M=M_1 \# M_2$ implies $M_1 = S^3$ or $M_2 = S^3$.<br />
A manifold $M$ is called irreducible if every embedded $S^2$ bounds a ball, i.e. the embedding extends to an embedding of $D^3$<br />
{{endthm}}<br />
Irreducibility is only slightly stronger than being prime. A orientable prime 3-manifold is either $S^2 \times S^1$ or every embedded 2-sphere bounds a ball.<br />
<br />
\begin{theorem}[Kneser]<br />
Every orientable, compact 3-manifold $M$ has a decomposition $M=P_1 \# \ldots \# P_n$ into prime manifolds $P_i$ unique up to ordering and $S^3$ summands.<br />
\end{theorem} <br />
<br />
A orientable prime 3-manifold is either $S^2 \times S^1$ or every embedded 2-sphere bounds a ball, in which case the manifold is called irreducible.<br />
<br />
Van Kampen's theorem tells you, that $\pi_1(M \# N)=\pi_1(M)*\pi_1(N)$. Hence any 3-manifold, whose fundamental group cannot be written as a free product of two nontrivial subgroups, can only be written as the connected sum of another 3-manifold with a simply connected 3-manifold. By the Poincaré conjecture a simply connected 3-manifold is already homeomorphic to $S^3$. Hence each such manifold is prime.<br />
<br />
Prime 3-manifolds can be distinguished by their fundamental groups into the following 3 types:<br />
</wikitex><br />
==== Type I: finite fundamental group==== <br />
<wikitex>;<br />
The universal cover $\tilde{M}$ is a simply-connected 3-manifold. As the fundamental group already determines the homology of a oriented, closed compact 3-manifold, it has to be a homology sphere. Using the Hurewicz-theorem, its fundamental class is represented by a degree 1 map $S^3 \rightarrow \tilde{M}$. This map induces isomorphisms on the homology and on the fundamental group. Hence it is a weak homotopy equivalence, and hence a homotopy equivalence by Whitehead's theorem (ref?).<br />
Hence every prime $3$-manifold with finite fundamental group arises as the quotient of a homotopy sphere by a free action of a finite group.<br />
With the use of the Poincaré conjecture every homotopy 3-sphere is homeomorphic to $S^3$ and we can write $M=S^3/\Gamma$. If $\Gamma$ is cyclic $M$ is known as lens space (ref). <br />
</wikitex><br />
==== Type II: infinite cyclic fundamental group ====<br />
<wikitex>;<br />
$S^1\times S^2$ is the only orientable closed prime 3-manifold of this type. Futhermore it is the only not irreducible prime manifold. (TODO: proof/ref)<br />
</wikitex><br />
<br />
==== Type III: infinite non-cyclic fundamental group ====<br />
<wikitex>;<br />
Such a manifold $M$ is always aspherical (TODO ref). The sphere theorem states, that every map $S^2\rightarrow M$ is homotopic to an embedding; and - as $M$ is irreducible - it is nullhomotopic. Hence $\pi_2(M)=0$. Consider the universal covering $\tilde{M}$ of $M$. Its first homology vanishes as it is simply connected. The long exact sequence of homotopy groups of the fibration $pi_1(M)\rightarrow \tilde{M}\rightarrow M$ gives a isomorphism $\pi_2(M)\cong \pi_2(\tilde{M})$. Hence by Hurewicz' theorem $H_2(\tilde{M})=0$. Furthermore $H_3(\tilde{M})=0$, as $M$ is noncompact. Applying Hurewicz theorem again we get that all homotopy groups of $\tilde{M}$ vanish and hence by Whitehead's theorem $\tilde{M}$ is contractible. This means that $M$ is apherical. <br />
Hence the homotopy type of a prime 3-manifold with infinite non-cyclic fundamental group is uniquely determined by its fundamental group.<br />
Furthermore not every group can occur as a fundamental group of a prime 3-manifold. The equivariant cellular chain complex of $\tilde{M}$ is a projective resolution of the trivial $\Zz[\pi_1(M)]$-module $\Zz$. Hence ....<br />
For any subgroup $F\le \pi_1(M)$ the space $\tilde{M}/F$ is a finite-dimensional model for $K(F,1)$. For example a finite group cannot have such a model (by group homology ref) and hence $\pi_1(M)$ must be torsionfree. Furthermore it is a Poincaré duality group (link).<br />
</wikitex><br />
<br />
=== Torus decomposition ===<br />
<wikitex>;<br />
According to the previous section it remains to classify irreducible prime 3-manifolds. <br />
After cutting along spheres which don't bound balls as far as possible the next canonical step is to consider incompressible tori which are disjoint from the boundary. <br />
<br />
\begin{theorem}[Jacob-Shalen, Johannson]<br />
If $M$ is an irreducible compact orientable manifold, then there is a collection of disjoint incompressible tori $T_1, \ldots ,T_n$ in $M$ such that splitting $M$ along the union of these tori produces manifolds $M_i$ which are either [[Wikipedia:Seifert_fiber_space|Seifert-fibered]] or atoroidal, i.e. every incompressible torus in $M_i$ is isotopic to a torus component of $\partial M_i$. Furthermore, a minimal such collection of tori $T_j$ is unique up to isotopy in $M$.<br />
\end{theorem}<br />
<br />
Thurston's geometrization conjectures states that all the pieces we get by this JSJ-decomposition admit one of eight possible geometric structures:<br />
There is a list of eight simply connected Riemannian manifolds - the so called model geometries. A geometric structure on $M$ is the choice of a Riemannian metric on $M$, with the property that its universal covering $ \tilde{M}$ equipped with the pull-back metric is isometric to one of the eight model geometries. It might a priori be easier to classify all cocompact actions on the several model geometries.<br />
<br />
The Seifert-fibered pieces are well understood since the work of Seifert in the 30s (TODO: mention classification theorem). The atoroidal pieces are described by the following Hyperbolization theorem which was stated by Thurston (ref) and proven by Perelman.<br />
\begin{thm}<br />
Every irreducible atoroidal closed 3-manifold that is not Seifert-fibred is hyperbolic.<br />
\end{thm}<br />
</wikitex><br />
<br />
=== Dehn surgery ===<br />
<wikitex><br />
Dehn surgery is a way of constructing (TODO oriented ? neccesary) 3-manifolds. Given a [[Wikipedia:Link_%28knot_theory%29 | link]] in a $3$-manifold $N$<br />
$$L: \coprod_{i=1}^n S^1\rightarrow N,$$<br />
and a choice of a tubular neighborhood of $L$ <br />
$$L': \coprod_{i=1}^n S^1\times D^2\rightarrow N\mbox{ with }L'(x,0)=L(x)$$.<br />
(This choice essentially is the choice of a trivialization of the normal bundle; TODO find a correct formulation for this).<br />
This gives us a family of embedded, disjoint, full tori. The idea of Dehn surgery is to remove these Tori and glue them back in using a twist.\\<br />
Let us restrict to the case with only one solid torus $L':S^1\times D^2\rightarrow N$. <br />
Choose any self-homeomorphism $f$ of the torus $S^1\times S^1$. The result of the Dehn surgery at $L$ with the twist $f$ is defined as <br />
$$N_{f,L'}:=N\setminus L'(S^1\times \mathring{D}^2) \cup_f S^1\times D^2=N\setminus L'(S^1\times \mathring{D}^2) \amalg S^1\times D^2/\sim,$$<br />
where the equivalence relation identifies for $(x,y)\in S^1\times S^1$ the points $L'(x,y)$ in the left component and $f(x,y)$ in the right component.<br />
If $f$ is the coordinate flipping, Dehn surgery is nothing but usual codimension $2$ surgery.<br />
<br />
\begin{lemma}<br />
Suppose $f,f'\in \Homeo(T^2)$ are isotopic and let $L':S^1\times D^2 \rightarrow N$ be any embedding of the full Torus in a $3$-Manifold $N$.Then $N_{f,L'}$ and $N_{f',L'}$ are homeomorphic.<br />
\end{lemma}<br />
\begin{proof}<br />
Let $j:T^2\times [0;1] \rightarrow T^2$ be an isotopy from $f$ to $f'$. This gives a homeomorphism:<br />
$$ J:T^2\times [0;1]\rightarrow T^2\times [0;1] \qquad (x,y,t)\mapsto (j(x,y,t),t). $$<br />
(TODO: is its inverse $(x,y,t)\mapsto (j(-,-,t)^{-1}(x,y),t)$ continuous ?).<br />
The idea is to grab some additional space, where one can use the map $J$. TODO<br />
\end{proof}<br />
TODO formulate a lemma, that M_{f,L'} also only depends on the isotopy class of $L'$ (which is hopefully true).<br />
Hence, we have to classify all self-homeomorphisms of $T^2$ up to isotopy.<br />
\begin{lemma}<br />
Every self-homeomorphism of $T^2$ is isotopic to exactly one homeomorphism of the shape <br />
$$f_A:\Rr^2/\Zz^2 \rightarrow \Rr^2 /\Zz^2 \qquad \left(\begin{array}{c}x\\y\end{array}\right)\mapsto A\cdot\left(\begin{array}{c}x\\y\end{array}\right),$$<br />
where $A\in GL_2(\Zz)$ (reference of proof). <br />
\end{lemma}<br />
\begin{proof} Since the torus is a $K\left(\Zz^2,1\right)$-space, we have that $\pi_0 Map\left(T^2,T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$ is an isomorphism. Homotopic surface homeomorphisms are isotopic (Reference?). Thus the restriction $\pi_0Homeo\left(T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$ is injective. Moreover, each $A\in Hom\left(\Zz^2,\Zz^2\right)$ is realised by $f_A$, therefore $\pi_0Homeo\left(T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$ is also surjective. <br />
TODO find a reference in ANY source about the mapping class group<br />
\end{proof}<br />
The next lemma tells us, that composition of self-homeomorphisms corresponds to two successive Dehn surgeries.<br />
\begin{lemma}<br />
Let $f,g\in \Homeo(T^2)$ be given and let $L':S^1\times D^2\rightarrow N$ is an embedding of the full torus in a 3-manifold. Then we have map <br />
$$L'':S^1 \times D^2 \rightarrow N\setminus L'(S^1\times \mathring{D}^2) \cup_f S^1\times D^2=N_{f,L'}$$<br />
given by the map $S^1\times D^2\rightarrow S^1 \times D^2 \quad (x,y)(x,y/2)$ postcomposed with the canonical inclusion in the second coordinate.<br />
Then $(N_{f,L'})_{g,L''} \cong N_{f\circ g,L'}$. TODO right order of composition ? We will see in the proof.<br />
\end{lemma}<br />
\beg{proof}<br />
\end{proof}<br />
<br />
We have to find out, which self-homeomorphisms of the torus don't change the homeomorphism type of the manifold.<br />
\begin{lemma} Consider a matrix of the form $\left( \begin{array}{cc} 1 & 0 \\ k & 1 \end{array}\right)$ and let $L':S^1\times D^2 \rightarrow N$ be any eembedding. Then $N_{f_A,L'}\cong N$.<br />
\end{lemma}<br />
TODO are there any orientation reversing homeos, that also extend ? Think so. Also add them here.<br />
\begin{proof}<br />
The homeomorphism $f_A \in \Homeo(T^2)$ extends to a homeomorphism of $\bar{f_A}\in \Homeo(S^1\times D^2)$:<br />
$$\bar{f_A}(x,y):=(x,x^ky), $$<br />
where $x\in S^1 = \{z\in \Cc| |z|=1\}, y\in D^2=\{y\in\Cc||y|\le 1\}$.<br />
Using this homeomorphism one can define a homeomorphism from $N_f$ to $N$:<br />
$$N = N\setminus L(S^1\times \mathring{D}^2)\cup_1 S^1\times D^2 \rightarrow N_{f_A}=N\setminus L(S^1 \times \mathring{D}^2)\cup_{f_A} S^1\times D^2$$<br />
given by the identity on the left component and $\bar{f_A}$ on the right component.<br />
\end{proof}<br />
<br />
Together with (link to comment about composition), this tells us, that $N_{f_A}$ really only depends on the coset $A\cdot \left(\begin{array}{cc}1&*\\0&1\end{array}\right)$ (TODO check right or left coset). This coset is uniquely determined by the image $(p,q)$ of $(1,0)$ with $p$ and $q$ coprime.<br />
<br />
The ratio $p/q$ is called the surgery coefficient. (TODO what is the quotient good for ?)<++><br />
<br />
<br />
TODO does the result give different manifolds.<br />
<br />
TODO does the result only depend on the isotopy class of the link. <br />
<br />
Every compact (oriented /able, neccesary ?) 3-manifold might be obtained from $S^3$ by a Dehn surgery along a link (TODO ref). Of course this does not satisfy to classify 3-manifolds without having a good classification of links in $S^3$. <br />
<br />
<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
\cite{Scott1983}, \cite{Thurston1997}, \cite{Hatcher2000}, \cite{Hempel1976}<br />
{{#RefList:}}<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_Euclidean_spacesGroup actions on Euclidean spaces2010-11-27T15:02:15Z<p>Marek Kaluba: /* Definitions */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
--><br />
{{Stub}}<br />
== Topological actions ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex><br />
The question whether contractible manifolds (e.g. Euclidean spaces) admit smooth fixed point free actions of compact Lie groups has been discussed for the first time by Paul Althaus Smith \cite{Smith1938}, \cite{Smith1939}, \cite{Smith1941}, and \cite{Smith1945}. Conner and Montgomery \cite{Conner&Montgomery1962} have constructed smooth fixed point free actions of $G$ on Euclidean spaces, for $G=SO(3)$. By generalizing their construction, Hsiang and Hsiang \cite{Hsiang&Hsiang1967} have shown that any non-abelian compact connected Lie group admits such actions. <br />
<br />
For $G=\mathbb{Z}_{pq}$ for two relatively primes integers $p,q\geq 2$, the construction of Conner and Floyd \cite{Conner&Floyd1959}, modified and improved by Kister \cite{Kister1961} and \cite{Kister1963}, yields a smooth fixed point free actions on Euclidean spaces (see {{cite|Bredon1972|pp. 58-61}}). For more general groups $G$ such that there exist a surjection $G\to\mathbb{Z}_p$ and an injection $\mathbb{Z}_q\to G$, such actions have been obtained by Edmonds and Lee \cite{Edmonds&Lee1976}.<br />
<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex><br />
The results of \cite{Conner&Montgomery1962}, \cite{Hsiang&Hsiang1967}, \cite{Conner&Floyd1959}, \cite{Kister1961}, and \cite{Edmonds&Lee1976} yield the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has smooth fixed point free action on some Euclidean space if and only if at least one of the following conditions holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian.<br />
*The quotient $G/G_0$ is not of prime power order.<br />
{{endthm}}<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternion $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$:<br />
$$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$ and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ <br />
{{beginrem|Definition}}<br />
For an abelian group $A$, $\operatorname{qDiv}(A)$ is the subgroup of quasi divisible elements of $A$, i.e., $\operatorname{qDiv}(A)$ is the intersection of the kernels of all homomorphisms from $A$ to free abelian group.<br />
{{endrem}}<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
<br />
{{beginthm|Theorem|(\cite{citation needed})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite dimensional CW-complex. Then the following two statements are equivalent.<br />
*$F$ consists of countable many cells.<br />
*There exist a finite dimensional, contractible $G$-CW-complex $X$ with finitely many orbit types, such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on some Euclidean space $E$ such that the fixed point set $E^G$ is homotopy equivalent to $F$.<br />
{{endthm}}<br />
<br />
It is assumed here, that any smooth manifold admits a countable smooth atlas.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1996})}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a smooth manifold. Then there exists a smooth action of $G$ on some Euclidean space $E$ such that the fixed point $E^G$ is diffeomorphic to $F$ if and only if the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies the following condition depending on $G$.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{qDiv}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{qDiv}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_Euclidean_spacesGroup actions on Euclidean spaces2010-11-27T15:00:41Z<p>Marek Kaluba: /* Definitions */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
--><br />
{{Stub}}<br />
== Topological actions ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex><br />
The question whether contractible manifolds (e.g. Euclidean spaces) admit smooth fixed point free actions of compact Lie groups has been discussed for the first time by Paul Althaus Smith \cite{Smith1938}, \cite{Smith1939}, \cite{Smith1941}, and \cite{Smith1945}. Conner and Montgomery \cite{Conner&Montgomery1962} have constructed smooth fixed point free actions of $G$ on Euclidean spaces, for $G=SO(3)$. By generalizing their construction, Hsiang and Hsiang \cite{Hsiang&Hsiang1967} have shown that any non-abelian compact connected Lie group admits such actions. <br />
<br />
For $G=\mathbb{Z}_{pq}$ for two relatively primes integers $p,q\geq 2$, the construction of Conner and Floyd \cite{Conner&Floyd1959}, modified and improved by Kister \cite{Kister1961} and \cite{Kister1963}, yields a smooth fixed point free actions on Euclidean spaces (see {{cite|Bredon1972|pp. 58-61}}). For more general groups $G$ such that there exist a surjection $G\to\mathbb{Z}_p$ and an injection $\mathbb{Z}_q\to G$, such actions have been obtained by Edmonds and Lee \cite{Edmonds&Lee1976}.<br />
<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex><br />
The results of \cite{Conner&Montgomery1962}, \cite{Hsiang&Hsiang1967}, \cite{Conner&Floyd1959}, \cite{Kister1961}, and \cite{Edmonds&Lee1976} yield the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has smooth fixed point free action on some Euclidean space if and only if at least one of the following conditions holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian.<br />
*The quotient $G/G_0$ is not of prime power order.<br />
{{endthm}}<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternion $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$:<br />
$$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$ and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ <br />
{{beginrem|Definition}}<br />
For a abelian group $A$, denote by $\operatorname{qDiv}(A)$ subgroup of quasi divisible elements of $A$, i.e., $\operatorname{qDiv}(A)$ is the intersection of the kernels of all homomorphisms from $A$ to free abelian groups.<br />
{{endrem}}<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
<br />
{{beginthm|Theorem|(\cite{citation needed})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite dimensional CW-complex. Then the following two statements are equivalent.<br />
*$F$ consists of countable many cells.<br />
*There exist a finite dimensional, contractible $G$-CW-complex $X$ with finitely many orbit types, such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on some Euclidean space $E$ such that the fixed point set $E^G$ is homotopy equivalent to $F$.<br />
{{endthm}}<br />
<br />
It is assumed here, that any smooth manifold admits a countable smooth atlas.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1996})}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a smooth manifold. Then there exists a smooth action of $G$ on some Euclidean space $E$ such that the fixed point $E^G$ is diffeomorphic to $F$ if and only if the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies the following condition depending on $G$.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{qDiv}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{qDiv}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_Euclidean_spacesGroup actions on Euclidean spaces2010-11-27T14:56:51Z<p>Marek Kaluba: /* Results */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
--><br />
{{Stub}}<br />
== Topological actions ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex><br />
The question whether contractible manifolds (e.g. Euclidean spaces) admit smooth fixed point free actions of compact Lie groups has been discussed for the first time by Paul Althaus Smith \cite{Smith1938}, \cite{Smith1939}, \cite{Smith1941}, and \cite{Smith1945}. Conner and Montgomery \cite{Conner&Montgomery1962} have constructed smooth fixed point free actions of $G$ on Euclidean spaces, for $G=SO(3)$. By generalizing their construction, Hsiang and Hsiang \cite{Hsiang&Hsiang1967} have shown that any non-abelian compact connected Lie group admits such actions. <br />
<br />
For $G=\mathbb{Z}_{pq}$ for two relatively primes integers $p,q\geq 2$, the construction of Conner and Floyd \cite{Conner&Floyd1959}, modified and improved by Kister \cite{Kister1961} and \cite{Kister1963}, yields a smooth fixed point free actions on Euclidean spaces (see {{cite|Bredon1972|pp. 58-61}}). For more general groups $G$ such that there exist a surjection $G\to\mathbb{Z}_p$ and an injection $\mathbb{Z}_q\to G$, such actions have been obtained by Edmonds and Lee \cite{Edmonds&Lee1976}.<br />
<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex><br />
The results of \cite{Conner&Montgomery1962}, \cite{Hsiang&Hsiang1967}, \cite{Conner&Floyd1959}, \cite{Kister1961}, and \cite{Edmonds&Lee1976} yield the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has smooth fixed point free action on some Euclidean space if and only if at least one of the following conditions holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian.<br />
*The quotient $G/G_0$ is not of prime power order.<br />
{{endthm}}<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternion $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$:<br />
$$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$ and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ <br />
<br />
For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$. <br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
<br />
{{beginthm|Theorem|(\cite{citation needed})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite dimensional CW-complex. Then the following two statements are equivalent.<br />
*$F$ consists of countable many cells.<br />
*There exist a finite dimensional, contractible $G$-CW-complex $X$ with finitely many orbit types, such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on some Euclidean space $E$ such that the fixed point set $E^G$ is homotopy equivalent to $F$.<br />
{{endthm}}<br />
<br />
It is assumed here, that any smooth manifold admits a countable smooth atlas.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1996})}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a smooth manifold. Then there exists a smooth action of $G$ on some Euclidean space $E$ such that the fixed point $E^G$ is diffeomorphic to $F$ if and only if the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies the following condition depending on $G$.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{qDiv}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{qDiv}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_Euclidean_spacesGroup actions on Euclidean spaces2010-11-27T14:49:46Z<p>Marek Kaluba: /* Results */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
--><br />
{{Stub}}<br />
== Topological actions ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex><br />
The question whether contractible manifolds (e.g. Euclidean spaces) admit smooth fixed point free actions of compact Lie groups has been discussed for the first time by Paul Althaus Smith \cite{Smith1938}, \cite{Smith1939}, \cite{Smith1941}, and \cite{Smith1945}. Conner and Montgomery \cite{Conner&Montgomery1962} have constructed smooth fixed point free actions of $G$ on Euclidean spaces, for $G=SO(3)$. By generalizing their construction, Hsiang and Hsiang \cite{Hsiang&Hsiang1967} have shown that any non-abelian compact connected Lie group admits such actions. <br />
<br />
For $G=\mathbb{Z}_{pq}$ for two relatively primes integers $p,q\geq 2$, the construction of Conner and Floyd \cite{Conner&Floyd1959}, modified and improved by Kister \cite{Kister1961} and \cite{Kister1963}, yields a smooth fixed point free actions on Euclidean spaces (see {{cite|Bredon1972|pp. 58-61}}). For more general groups $G$ such that there exist a surjection $G\to\mathbb{Z}_p$ and an injection $\mathbb{Z}_q\to G$, such actions have been obtained by Edmonds and Lee \cite{Edmonds&Lee1976}.<br />
<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex><br />
The results of \cite{Conner&Montgomery1962}, \cite{Hsiang&Hsiang1967}, \cite{Conner&Floyd1959}, \cite{Kister1961}, and \cite{Edmonds&Lee1976} yield the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has smooth fixed point free action on some Euclidean space if and only if at least one of the following conditions holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian.<br />
*The quotient $G/G_0$ is not of prime power order.<br />
{{endthm}}<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternion $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$:<br />
$$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$ and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ <br />
<br />
For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$. <br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1975}, \cite{Oliver1976})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite dimensional, with countable many cells CW-complex. Then the following two statements are equivalent.<br />
*There exist a finite dimensional, contractible $G$-CW-complex $X$ with finitely many orbit types, such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on some Euclidean space $E$ such that the fixed point set $E^G$ is homotopy equivalent to $F$.<br />
{{endthm}}<br />
<br />
It is assumed here, that any smooth manifold admits a countable smooth atlas.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1996})}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a smooth manifold. Then there exists a smooth action of $G$ on some Euclidean space $E$ such that the fixed point $E^G$ is diffeomorphic to $F$ if and only if the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies the following condition depending on $G$.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{qDiv}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{qDiv}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_Euclidean_spacesGroup actions on Euclidean spaces2010-11-27T14:48:17Z<p>Marek Kaluba: /* Results */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
--><br />
{{Stub}}<br />
== Topological actions ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex><br />
The question whether contractible manifolds (e.g. Euclidean spaces) admit smooth fixed point free actions of compact Lie groups has been discussed for the first time by Paul Althaus Smith \cite{Smith1938}, \cite{Smith1939}, \cite{Smith1941}, and \cite{Smith1945}. Conner and Montgomery \cite{Conner&Montgomery1962} have constructed smooth fixed point free actions of $G$ on Euclidean spaces, for $G=SO(3)$. By generalizing their construction, Hsiang and Hsiang \cite{Hsiang&Hsiang1967} have shown that any non-abelian compact connected Lie group admits such actions. <br />
<br />
For $G=\mathbb{Z}_{pq}$ for two relatively primes integers $p,q\geq 2$, the construction of Conner and Floyd \cite{Conner&Floyd1959}, modified and improved by Kister \cite{Kister1961} and \cite{Kister1963}, yields a smooth fixed point free actions on Euclidean spaces (see {{cite|Bredon1972|pp. 58-61}}). For more general groups $G$ such that there exist a surjection $G\to\mathbb{Z}_p$ and an injection $\mathbb{Z}_q\to G$, such actions have been obtained by Edmonds and Lee \cite{Edmonds&Lee1976}.<br />
<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex><br />
The results of \cite{Conner&Montgomery1962}, \cite{Hsiang&Hsiang1967}, \cite{Conner&Floyd1959}, \cite{Kister1961}, and \cite{Edmonds&Lee1976} yield the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has smooth fixed point free action on some Euclidean space if and only if at least one of the following conditions holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian.<br />
*The quotient $G/G_0$ is not of prime power order.<br />
{{endthm}}<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternion $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$:<br />
$$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$ and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ <br />
<br />
For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$. <br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1975}, \cite{Oliver1976})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite dimensional, with countable many cells CW-complex. Then the following two statements are equivalent.<br />
*There exist a finite dimensional, contractible $G$-CW-complex $X$ with finitely many orbit types, such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on some Euclidean space $E$ such that the fixed point set $E^G$ is homotopy equivalent to $F$.<br />
{{endthm}}<br />
<br />
Any smooth manifold considered here is second countable (i.e. admits a countable smooth atlas).<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1996})}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a smooth manifold. Then there exists a smooth action of $G$ on some Euclidean space $E$ such that the fixed point $E^G$ is diffeomorphic to $F$ if and only if the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies the following condition depending on $G$.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{qDiv}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{qDiv}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_disksGroup actions on disks2010-11-27T14:38:42Z<p>Marek Kaluba: /* Fixed point sets */</p>
<hr />
<div>{{Stub}}<br />
== Topological actions ==<br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex>;<br />
Floyd and Richardson \cite{Floyd&Richardson1959} have constructed for the first time a smooth fixed point free action of $G$ on a disk for $G=A_5$, the alternating group on five letters (see {{cite|Bredon1972|pp. 55-58}} for a transparent description of the construction). Next, Greever {{cite|Greever1960}} has described plenty of finite solvable groups $G$, which can act smoothly on disks without fixed points. Then, Oliver \cite{Oliver1975} and \cite{Oliver1976}, has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks. <br />
</wikitex><br />
<br />
==== Oliver number ====<br />
<wikitex>;<br />
Let $G$ be a finite group not of prime power order. Oliver in \cite{Oliver1975} proved that the set $$\{\chi(X^G)-1\colon X\textup{ is a contractible }G\textup{-CW-complex}\}$$ is a subgroup of the group of integers $\mathbb{Z}$. Therefore, the set is of the form $n_G\cdot \mathbb{Z}$ for a unique integer $n_G\geq 0$, which we refer to as the ''Oliver number'' of $G$. <br />
<br />
Oliver has determined integer $n_G$ in the papers \cite{Oliver1975}, \cite{Oliver1977}, and \cite{Oliver1978}. In particular, the following lemma holds.<br />
<br />
{{beginthm|Lemma|(Oliver Lemma)}}<br />
For a finite group $G$ not of prime power order, $n_G=1$ if and only if there does not exist a sequence $P\trianglelefteq H\trianglelefteq G$ of normal subgroups such that $P$ is a $p$-group, $G/H$ is a $q$-group, and $H/P$ is cyclic for two (possibly the same) primes $p$ and $q$.<br />
{{endthm}}<br />
<br />
Moreover, the work {{cite|Oliver1977|Theorem 7}} yields the following proposition.<br />
<br />
{{beginthm|Proposition}}<br />
For a finite nilpotent group $G$ not of prime power order, the following conclusions hold:<br />
<br />
*$n_G=0$ if $G$ has at most one non-cyclic Sylow subgroup.<br />
*$n_G=pq$ for two distinct primes $p$ and $q$, if $G$ has just one non-cyclic $p$-Sylow and $q$-Sylow subgroups.<br />
*$n_G=1$ if $G$ has three or more non-cyclic Sylow subgroups.<br />
<br />
{{endthm}}<br />
<br />
The notion of the Oliver number $n_G$ extends to compact Lie groups $G$ as follows.<br />
<br />
*$n_G=n_{G/G_0}$ if $G_0$ is abelian and $G/G_0$ is not of prime power order.<br />
*$n_G=1$ if $G_0$ is non-abelian (see \cite{Oliver1976}).<br />
<br />
</wikitex><br />
<br />
==== Oliver group ====<br />
<wikitex>;<br />
In connection with the work on [[Group_actions_on_spheres#Smooth_actions|smooth one fixed point actions on spheres]], Laitinen and Morimoto \cite{Laitinen&Morimoto1998} have introduced the notion of Oliver group. <br />
{{beginrem|Definition}}<br />
A finite group $G$ not of prime power order is called an ''Oliver group'' if $n_G=1$.<br />
{{endrem}}<br />
<br />
Examples of finite Oliver groups include:<br />
*$\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ for three distinct primes $p$, $q$, and $r$.<br />
*the groups $S_4\oplus \mathbb{Z}_3$ and $A_4\oplus S_3$ of order 72 (which are solvable but not nilpotent).<br />
*finite non-solvable (in particular, non-trivial perfect) groups (e.g. $A_n$, $S_n$ for $n\geq 5$).<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
The results of Oliver \cite{Oliver1975} and \cite{Oliver1976}, are summarized in the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has a smooth fixed point free action on some disk if and only if at least one of the following condition holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian. <br />
*The quotient $G/G_0$ is not of prime power order and $n_{G/G_0}=1$.<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternion $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$:<br />
$$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$ and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ <br />
<br />
For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$. <br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1975}, \cite{Oliver1976})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite CW-complex. Then the following three statements are equivalent.<br />
*The Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$.<br />
*There exist a finite contractible $G$-CW-complex $X$ such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on a disk $D$ such that the fixed point set $D^G$ is homotopy equivalent to $F$.<br />
<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1996})}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a compact smooth manifold. Then there exists a smooth action of $G$ on some disk $D$ such that the fixed point $D^G$ is diffeomorphic to $F$ if and only if $\chi(F)\equiv 1\pmod{n_G}$ and the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies the following condition depending on $G.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_Euclidean_spacesGroup actions on Euclidean spaces2010-11-27T14:37:30Z<p>Marek Kaluba: /* Results */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
--><br />
{{Stub}}<br />
== Topological actions ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex><br />
The question whether contractible manifolds (e.g. Euclidean spaces) admit smooth fixed point free actions of compact Lie groups has been discussed for the first time by Paul Althaus Smith \cite{Smith1938}, \cite{Smith1939}, \cite{Smith1941}, and \cite{Smith1945}. Conner and Montgomery \cite{Conner&Montgomery1962} have constructed smooth fixed point free actions of $G$ on Euclidean spaces, for $G=SO(3)$. By generalizing their construction, Hsiang and Hsiang \cite{Hsiang&Hsiang1967} have shown that any non-abelian compact connected Lie group admits such actions. <br />
<br />
For $G=\mathbb{Z}_{pq}$ for two relatively primes integers $p,q\geq 2$, the construction of Conner and Floyd \cite{Conner&Floyd1959}, modified and improved by Kister \cite{Kister1961} and \cite{Kister1963}, yields a smooth fixed point free actions on Euclidean spaces (see {{cite|Bredon1972|pp. 58-61}}). For more general groups $G$ such that there exist a surjection $G\to\mathbb{Z}_p$ and an injection $\mathbb{Z}_q\to G$, such actions have been obtained by Edmonds and Lee \cite{Edmonds&Lee1976}.<br />
<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex><br />
The results of \cite{Conner&Montgomery1962}, \cite{Hsiang&Hsiang1967}, \cite{Conner&Floyd1959}, \cite{Kister1961}, and \cite{Edmonds&Lee1976} yield the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has smooth fixed point free action on some Euclidean space if and only if at least one of the following conditions holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian.<br />
*The quotient $G/G_0$ is not of prime power order.<br />
{{endthm}}<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternion $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$:<br />
$$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$ and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ <br />
<br />
For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$. <br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1975}, \cite{Oliver1976})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite CW-complex. Then the following three statements are equivalent.<br />
*The Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$.<br />
*There exist a finite contractible $G$-CW-complex $X$ such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on a disk $D$ such that the fixed point set $D^G$ is homotopy equivalent to $F$.<br />
<br />
{{endthm}}<br />
<br />
By smooth ma<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1996})}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a smooth manifold. Then there exists a smooth action of $G$ on some Euclidean space $E$ such that the fixed point $E^G$ is diffeomorphic to $F$ if and only if the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies the following condition depending on $G$.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{qDiv}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{qDiv}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_Euclidean_spacesGroup actions on Euclidean spaces2010-11-27T14:36:50Z<p>Marek Kaluba: /* Results */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
--><br />
{{Stub}}<br />
== Topological actions ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex><br />
The question whether contractible manifolds (e.g. Euclidean spaces) admit smooth fixed point free actions of compact Lie groups has been discussed for the first time by Paul Althaus Smith \cite{Smith1938}, \cite{Smith1939}, \cite{Smith1941}, and \cite{Smith1945}. Conner and Montgomery \cite{Conner&Montgomery1962} have constructed smooth fixed point free actions of $G$ on Euclidean spaces, for $G=SO(3)$. By generalizing their construction, Hsiang and Hsiang \cite{Hsiang&Hsiang1967} have shown that any non-abelian compact connected Lie group admits such actions. <br />
<br />
For $G=\mathbb{Z}_{pq}$ for two relatively primes integers $p,q\geq 2$, the construction of Conner and Floyd \cite{Conner&Floyd1959}, modified and improved by Kister \cite{Kister1961} and \cite{Kister1963}, yields a smooth fixed point free actions on Euclidean spaces (see {{cite|Bredon1972|pp. 58-61}}). For more general groups $G$ such that there exist a surjection $G\to\mathbb{Z}_p$ and an injection $\mathbb{Z}_q\to G$, such actions have been obtained by Edmonds and Lee \cite{Edmonds&Lee1976}.<br />
<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex><br />
The results of \cite{Conner&Montgomery1962}, \cite{Hsiang&Hsiang1967}, \cite{Conner&Floyd1959}, \cite{Kister1961}, and \cite{Edmonds&Lee1976} yield the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has smooth fixed point free action on some Euclidean space if and only if at least one of the following conditions holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian.<br />
*The quotient $G/G_0$ is not of prime power order.<br />
{{endthm}}<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternion $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$:<br />
$$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$ and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ <br />
<br />
For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$. <br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1975}, \cite{Oliver1976})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite CW-complex. Then the following three statements are equivalent.<br />
*The Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$.<br />
*There exist a finite contractible $G$-CW-complex $X$ such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on a disk $D$ such that the fixed point set $D^G$ is homotopy equivalent to $F$.<br />
<br />
{{endthm}}<br />
<br />
By smooth ma<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1996})}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a smooth manifold. Then there exists a smooth action of $G$ on some Euclidean space $E$ such that the fixed point $E^G$ is diffeomorphic to $F$ if and only if the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies the following condition depending o $G$.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{qDiv}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{qDiv}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_Euclidean_spacesGroup actions on Euclidean spaces2010-11-27T14:29:04Z<p>Marek Kaluba: /* Fixed point sets */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
--><br />
{{Stub}}<br />
== Topological actions ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex><br />
The question whether contractible manifolds (e.g. Euclidean spaces) admit smooth fixed point free actions of compact Lie groups has been discussed for the first time by Paul Althaus Smith \cite{Smith1938}, \cite{Smith1939}, \cite{Smith1941}, and \cite{Smith1945}. Conner and Montgomery \cite{Conner&Montgomery1962} have constructed smooth fixed point free actions of $G$ on Euclidean spaces, for $G=SO(3)$. By generalizing their construction, Hsiang and Hsiang \cite{Hsiang&Hsiang1967} have shown that any non-abelian compact connected Lie group admits such actions. <br />
<br />
For $G=\mathbb{Z}_{pq}$ for two relatively primes integers $p,q\geq 2$, the construction of Conner and Floyd \cite{Conner&Floyd1959}, modified and improved by Kister \cite{Kister1961} and \cite{Kister1963}, yields a smooth fixed point free actions on Euclidean spaces (see {{cite|Bredon1972|pp. 58-61}}). For more general groups $G$ such that there exist a surjection $G\to\mathbb{Z}_p$ and an injection $\mathbb{Z}_q\to G$, such actions have been obtained by Edmonds and Lee \cite{Edmonds&Lee1976}.<br />
<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex><br />
The results of \cite{Conner&Montgomery1962}, \cite{Hsiang&Hsiang1967}, \cite{Conner&Floyd1959}, \cite{Kister1961}, and \cite{Edmonds&Lee1976} yield the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has smooth fixed point free action on some Euclidean space if and only if at least one of the following conditions holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian.<br />
*The quotient $G/G_0$ is not of prime power order.<br />
{{endthm}}<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternion $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$:<br />
$$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$ and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ <br />
<br />
For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$. <br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1975}, \cite{Oliver1976})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite CW-complex. Then the following three statements are equivalent.<br />
*The Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$.<br />
*There exist a finite contractible $G$-CW-complex $X$ such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on a disk $D$ such that the fixed point set $D^G$ is homotopy equivalent to $F$.<br />
<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1996})}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a compact smooth manifold. Then there exists a smooth action of $G$ on some disk $D$ such that the fixed point $D^G$ is diffeomorphic to $F$ if and only if $\chi(F)\equiv 1\pmod{n_G}$ and the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies one of the following condition.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_disksGroup actions on disks2010-11-27T14:25:55Z<p>Marek Kaluba: /* Results */</p>
<hr />
<div>{{Stub}}<br />
== Topological actions ==<br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex>;<br />
Floyd and Richardson \cite{Floyd&Richardson1959} have constructed for the first time a smooth fixed point free action of $G$ on a disk for $G=A_5$, the alternating group on five letters (see {{cite|Bredon1972|pp. 55-58}} for a transparent description of the construction). Next, Greever {{cite|Greever1960}} has described plenty of finite solvable groups $G$, which can act smoothly on disks without fixed points. Then, Oliver \cite{Oliver1975} and \cite{Oliver1976}, has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks. <br />
</wikitex><br />
<br />
==== Oliver number ====<br />
<wikitex>;<br />
Let $G$ be a finite group not of prime power order. Oliver in \cite{Oliver1975} proved that the set $$\{\chi(X^G)-1\colon X\textup{ is a contractible }G\textup{-CW-complex}\}$$ is a subgroup of the group of integers $\mathbb{Z}$. Therefore, the set is of the form $n_G\cdot \mathbb{Z}$ for a unique integer $n_G\geq 0$, which we refer to as the ''Oliver number'' of $G$. <br />
<br />
Oliver has determined integer $n_G$ in the papers \cite{Oliver1975}, \cite{Oliver1977}, and \cite{Oliver1978}. In particular, the following lemma holds.<br />
<br />
{{beginthm|Lemma|(Oliver Lemma)}}<br />
For a finite group $G$ not of prime power order, $n_G=1$ if and only if there does not exist a sequence $P\trianglelefteq H\trianglelefteq G$ of normal subgroups such that $P$ is a $p$-group, $G/H$ is a $q$-group, and $H/P$ is cyclic for two (possibly the same) primes $p$ and $q$.<br />
{{endthm}}<br />
<br />
Moreover, the work {{cite|Oliver1977|Theorem 7}} yields the following proposition.<br />
<br />
{{beginthm|Proposition}}<br />
For a finite nilpotent group $G$ not of prime power order, the following conclusions hold:<br />
<br />
*$n_G=0$ if $G$ has at most one non-cyclic Sylow subgroup.<br />
*$n_G=pq$ for two distinct primes $p$ and $q$, if $G$ has just one non-cyclic $p$-Sylow and $q$-Sylow subgroups.<br />
*$n_G=1$ if $G$ has three or more non-cyclic Sylow subgroups.<br />
<br />
{{endthm}}<br />
<br />
The notion of the Oliver number $n_G$ extends to compact Lie groups $G$ as follows.<br />
<br />
*$n_G=n_{G/G_0}$ if $G_0$ is abelian and $G/G_0$ is not of prime power order.<br />
*$n_G=1$ if $G_0$ is non-abelian (see \cite{Oliver1976}).<br />
<br />
</wikitex><br />
<br />
==== Oliver group ====<br />
<wikitex>;<br />
In connection with the work on [[Group_actions_on_spheres#Smooth_actions|smooth one fixed point actions on spheres]], Laitinen and Morimoto \cite{Laitinen&Morimoto1998} have introduced the notion of Oliver group. <br />
{{beginrem|Definition}}<br />
A finite group $G$ not of prime power order is called an ''Oliver group'' if $n_G=1$.<br />
{{endrem}}<br />
<br />
Examples of finite Oliver groups include:<br />
*$\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ for three distinct primes $p$, $q$, and $r$.<br />
*the groups $S_4\oplus \mathbb{Z}_3$ and $A_4\oplus S_3$ of order 72 (which are solvable but not nilpotent).<br />
*finite non-solvable (in particular, non-trivial perfect) groups (e.g. $A_n$, $S_n$ for $n\geq 5$).<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
The results of Oliver \cite{Oliver1975} and \cite{Oliver1976}, are summarized in the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has a smooth fixed point free action on some disk if and only if at least one of the following condition holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian. <br />
*The quotient $G/G_0$ is not of prime power order and $n_{G/G_0}=1$.<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternion $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$:<br />
$$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$ and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ <br />
<br />
For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$. <br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1975}, \cite{Oliver1976})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite CW-complex. Then the following three statements are equivalent.<br />
*The Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$.<br />
*There exist a finite contractible $G$-CW-complex $X$ such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on a disk $D$ such that the fixed point set $D^G$ is homotopy equivalent to $F$.<br />
<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1996})}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a compact smooth manifold. Then there exists a smooth action of $G$ on some disk $D$ such that the fixed point $D^G$ is diffeomorphic to $F$ if and only if $\chi(F)\equiv 1\pmod{n_G}$ and the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies one of the following condition.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_disksGroup actions on disks2010-11-27T14:25:15Z<p>Marek Kaluba: /* Definitions */</p>
<hr />
<div>{{Stub}}<br />
== Topological actions ==<br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex>;<br />
Floyd and Richardson \cite{Floyd&Richardson1959} have constructed for the first time a smooth fixed point free action of $G$ on a disk for $G=A_5$, the alternating group on five letters (see {{cite|Bredon1972|pp. 55-58}} for a transparent description of the construction). Next, Greever {{cite|Greever1960}} has described plenty of finite solvable groups $G$, which can act smoothly on disks without fixed points. Then, Oliver \cite{Oliver1975} and \cite{Oliver1976}, has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks. <br />
</wikitex><br />
<br />
==== Oliver number ====<br />
<wikitex>;<br />
Let $G$ be a finite group not of prime power order. Oliver in \cite{Oliver1975} proved that the set $$\{\chi(X^G)-1\colon X\textup{ is a contractible }G\textup{-CW-complex}\}$$ is a subgroup of the group of integers $\mathbb{Z}$. Therefore, the set is of the form $n_G\cdot \mathbb{Z}$ for a unique integer $n_G\geq 0$, which we refer to as the ''Oliver number'' of $G$. <br />
<br />
Oliver has determined integer $n_G$ in the papers \cite{Oliver1975}, \cite{Oliver1977}, and \cite{Oliver1978}. In particular, the following lemma holds.<br />
<br />
{{beginthm|Lemma|(Oliver Lemma)}}<br />
For a finite group $G$ not of prime power order, $n_G=1$ if and only if there does not exist a sequence $P\trianglelefteq H\trianglelefteq G$ of normal subgroups such that $P$ is a $p$-group, $G/H$ is a $q$-group, and $H/P$ is cyclic for two (possibly the same) primes $p$ and $q$.<br />
{{endthm}}<br />
<br />
Moreover, the work {{cite|Oliver1977|Theorem 7}} yields the following proposition.<br />
<br />
{{beginthm|Proposition}}<br />
For a finite nilpotent group $G$ not of prime power order, the following conclusions hold:<br />
<br />
*$n_G=0$ if $G$ has at most one non-cyclic Sylow subgroup.<br />
*$n_G=pq$ for two distinct primes $p$ and $q$, if $G$ has just one non-cyclic $p$-Sylow and $q$-Sylow subgroups.<br />
*$n_G=1$ if $G$ has three or more non-cyclic Sylow subgroups.<br />
<br />
{{endthm}}<br />
<br />
The notion of the Oliver number $n_G$ extends to compact Lie groups $G$ as follows.<br />
<br />
*$n_G=n_{G/G_0}$ if $G_0$ is abelian and $G/G_0$ is not of prime power order.<br />
*$n_G=1$ if $G_0$ is non-abelian (see \cite{Oliver1976}).<br />
<br />
</wikitex><br />
<br />
==== Oliver group ====<br />
<wikitex>;<br />
In connection with the work on [[Group_actions_on_spheres#Smooth_actions|smooth one fixed point actions on spheres]], Laitinen and Morimoto \cite{Laitinen&Morimoto1998} have introduced the notion of Oliver group. <br />
{{beginrem|Definition}}<br />
A finite group $G$ not of prime power order is called an ''Oliver group'' if $n_G=1$.<br />
{{endrem}}<br />
<br />
Examples of finite Oliver groups include:<br />
*$\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ for three distinct primes $p$, $q$, and $r$.<br />
*the groups $S_4\oplus \mathbb{Z}_3$ and $A_4\oplus S_3$ of order 72 (which are solvable but not nilpotent).<br />
*finite non-solvable (in particular, non-trivial perfect) groups (e.g. $A_n$, $S_n$ for $n\geq 5$).<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
The results of Oliver \cite{Oliver1975} and \cite{Oliver1976}, are summarized in the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has a smooth fixed point free action on some disk if and only if at least one of the following condition holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian. <br />
*The quotient $G/G_0$ is not of prime power order and $n_{G/G_0}=1$.<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternion $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$:<br />
$$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$ and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ <br />
<br />
For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$. <br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1975},\cite{Oliver1976})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite CW-complex. Then the following three statements are equivalent.<br />
*The Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$.<br />
*There exist a finite contractible $G$-CW-complex $X$ such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on a disk $D$ such that the fixed point set $D^G$ is homotopy equivalent to $F$.<br />
<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem|\cite{Oliver1996}}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a compact smooth manifold. Then there exists a smooth action of $G$ on some disk $D$ such that the fixed point $D^G$ is diffeomorphic to $F$ if and only if $\chi(F)\equiv 1\pmod{n_G}$ and the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies one of the following condition.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_disksGroup actions on disks2010-11-27T14:24:34Z<p>Marek Kaluba: /* Definitions */</p>
<hr />
<div>{{Stub}}<br />
== Topological actions ==<br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex>;<br />
Floyd and Richardson \cite{Floyd&Richardson1959} have constructed for the first time a smooth fixed point free action of $G$ on a disk for $G=A_5$, the alternating group on five letters (see {{cite|Bredon1972|pp. 55-58}} for a transparent description of the construction). Next, Greever {{cite|Greever1960}} has described plenty of finite solvable groups $G$, which can act smoothly on disks without fixed points. Then, Oliver \cite{Oliver1975} and \cite{Oliver1976}, has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks. <br />
</wikitex><br />
<br />
==== Oliver number ====<br />
<wikitex>;<br />
Let $G$ be a finite group not of prime power order. Oliver in \cite{Oliver1975} proved that the set $$\{\chi(X^G)-1\colon X\textup{ is a contractible }G\textup{-CW-complex}\}$$ is a subgroup of the group of integers $\mathbb{Z}$. Therefore, the set is of the form $n_G\cdot \mathbb{Z}$ for a unique integer $n_G\geq 0$, which we refer to as the ''Oliver number'' of $G$. <br />
<br />
Oliver has determined integer $n_G$ in the papers \cite{Oliver1975}, \cite{Oliver1977}, and \cite{Oliver1978}. In particular, the following lemma holds.<br />
<br />
{{beginthm|Lemma|(Oliver Lemma)}}<br />
For a finite group $G$ not of prime power order, $n_G=1$ if and only if there does not exist a sequence $P\trianglelefteq H\trianglelefteq G$ of normal subgroups such that $P$ is a $p$-group, $G/H$ is a $q$-group, and $H/P$ is cyclic for two (possibly the same) primes $p$ and $q$.<br />
{{endthm}}<br />
<br />
Moreover, the work {{cite|Oliver1977|Theorem 7}} yields the following proposition.<br />
<br />
{{beginthm|Proposition}}<br />
For a finite nilpotent group $G$ not of prime power order, the following conclusions hold:<br />
<br />
*$n_G=0$ if $G$ has at most one non-cyclic Sylow subgroup.<br />
*$n_G=pq$ for two distinct primes $p$ and $q$, if $G$ has just one non-cyclic $p$-Sylow and $q$-Sylow subgroups.<br />
*$n_G=1$ if $G$ has three or more non-cyclic Sylow subgroups.<br />
<br />
{{endthm}}<br />
<br />
The notion of the Oliver number $n_G$ extends to compact Lie groups $G$ as follows.<br />
<br />
*$n_G=n_{G/G_0}$ if $G_0$ is abelian and $G/G_0$ is not of prime power order.<br />
*$n_G=1$ if $G_0$ is non-abelian (see \cite{Oliver1976}).<br />
<br />
</wikitex><br />
<br />
==== Oliver group ====<br />
<wikitex>;<br />
In connection with the work on [[Group_actions_on_spheres#Smooth_actions|smooth one fixed point actions on spheres]], Laitinen and Morimoto \cite{Laitinen&Morimoto1998} have introduced the notion of Oliver group. <br />
{{beginrem|Definition}}<br />
A finite group $G$ not of prime power order is called an ''Oliver group'' if $n_G=1$.<br />
{{endrem}}<br />
<br />
Examples of finite Oliver groups include:<br />
*$\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ for three distinct primes $p$, $q$, and $r$.<br />
*the groups $S_4\oplus \mathbb{Z}_3$ and $A_4\oplus S_3$ of order 72 (which are solvable but not nilpotent).<br />
*finite non-solvable (in particular, non-trivial perfect) groups (e.g. $A_n$, $S_n$ for $n\geq 5$).<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
The results of Oliver \cite{Oliver1975} and \cite{Oliver1976}, are summarized in the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has a smooth fixed point free action on some disk if and only if at least one of the following condition holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian. <br />
*The quotient $G/G_0$ is not of prime power order and $n_{G/G_0}=1$.<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternions $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$:<br />
$$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$ and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ <br />
<br />
For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$. <br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1975},\cite{Oliver1976})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite CW-complex. Then the following three statements are equivalent.<br />
*The Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$.<br />
*There exist a finite contractible $G$-CW-complex $X$ such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on a disk $D$ such that the fixed point set $D^G$ is homotopy equivalent to $F$.<br />
<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem|\cite{Oliver1996}}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a compact smooth manifold. Then there exists a smooth action of $G$ on some disk $D$ such that the fixed point $D^G$ is diffeomorphic to $F$ if and only if $\chi(F)\equiv 1\pmod{n_G}$ and the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies one of the following condition.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_disksGroup actions on disks2010-11-27T14:22:25Z<p>Marek Kaluba: /* Definitions */</p>
<hr />
<div>{{Stub}}<br />
== Topological actions ==<br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex>;<br />
Floyd and Richardson \cite{Floyd&Richardson1959} have constructed for the first time a smooth fixed point free action of $G$ on a disk for $G=A_5$, the alternating group on five letters (see {{cite|Bredon1972|pp. 55-58}} for a transparent description of the construction). Next, Greever {{cite|Greever1960}} has described plenty of finite solvable groups $G$, which can act smoothly on disks without fixed points. Then, Oliver \cite{Oliver1975} and \cite{Oliver1976}, has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks. <br />
</wikitex><br />
<br />
==== Oliver number ====<br />
<wikitex>;<br />
Let $G$ be a finite group not of prime power order. Oliver in \cite{Oliver1975} proved that the set $$\{\chi(X^G)-1\colon X\textup{ is a contractible }G\textup{-CW-complex}\}$$ is a subgroup of the group of integers $\mathbb{Z}$. Therefore, the set is of the form $n_G\cdot \mathbb{Z}$ for a unique integer $n_G\geq 0$, which we refer to as the ''Oliver number'' of $G$. <br />
<br />
Oliver has determined integer $n_G$ in the papers \cite{Oliver1975}, \cite{Oliver1977}, and \cite{Oliver1978}. In particular, the following lemma holds.<br />
<br />
{{beginthm|Lemma|(Oliver Lemma)}}<br />
For a finite group $G$ not of prime power order, $n_G=1$ if and only if there does not exist a sequence $P\trianglelefteq H\trianglelefteq G$ of normal subgroups such that $P$ is a $p$-group, $G/H$ is a $q$-group, and $H/P$ is cyclic for two (possibly the same) primes $p$ and $q$.<br />
{{endthm}}<br />
<br />
Moreover, the work {{cite|Oliver1977|Theorem 7}} yields the following proposition.<br />
<br />
{{beginthm|Proposition}}<br />
For a finite nilpotent group $G$ not of prime power order, the following conclusions hold:<br />
<br />
*$n_G=0$ if $G$ has at most one non-cyclic Sylow subgroup.<br />
*$n_G=pq$ for two distinct primes $p$ and $q$, if $G$ has just one non-cyclic $p$-Sylow and $q$-Sylow subgroups.<br />
*$n_G=1$ if $G$ has three or more non-cyclic Sylow subgroups.<br />
<br />
{{endthm}}<br />
<br />
The notion of the Oliver number $n_G$ extends to compact Lie groups $G$ as follows.<br />
<br />
*$n_G=n_{G/G_0}$ if $G_0$ is abelian and $G/G_0$ is not of prime power order.<br />
*$n_G=1$ if $G_0$ is non-abelian (see \cite{Oliver1976}).<br />
<br />
</wikitex><br />
<br />
==== Oliver group ====<br />
<wikitex>;<br />
In connection with the work on [[Group_actions_on_spheres#Smooth_actions|smooth one fixed point actions on spheres]], Laitinen and Morimoto \cite{Laitinen&Morimoto1998} have introduced the notion of Oliver group. <br />
{{beginrem|Definition}}<br />
A finite group $G$ not of prime power order is called an ''Oliver group'' if $n_G=1$.<br />
{{endrem}}<br />
<br />
Examples of finite Oliver groups include:<br />
*$\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ for three distinct primes $p$, $q$, and $r$.<br />
*the groups $S_4\oplus \mathbb{Z}_3$ and $A_4\oplus S_3$ of order 72 (which are solvable but not nilpotent).<br />
*finite non-solvable (in particular, non-trivial perfect) groups (e.g. $A_n$, $S_n$ for $n\geq 5$).<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
The results of Oliver \cite{Oliver1975} and \cite{Oliver1976}, are summarized in the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has a smooth fixed point free action on some disk if and only if at least one of the following condition holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian. <br />
*The quotient $G/G_0$ is not of prime power order and $n_{G/G_0}=1$.<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternions $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ and the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$:<br />
<br />
$$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F).$$<br />
<br />
For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$. <br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1975},\cite{Oliver1976})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite CW-complex. Then the following three statements are equivalent.<br />
*The Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$.<br />
*There exist a finite contractible $G$-CW-complex $X$ such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on a disk $D$ such that the fixed point set $D^G$ is homotopy equivalent to $F$.<br />
<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem|\cite{Oliver1996}}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a compact smooth manifold. Then there exists a smooth action of $G$ on some disk $D$ such that the fixed point $D^G$ is diffeomorphic to $F$ if and only if $\chi(F)\equiv 1\pmod{n_G}$ and the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies one of the following condition.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_disksGroup actions on disks2010-11-27T14:15:35Z<p>Marek Kaluba: /* Definitions */</p>
<hr />
<div>{{Stub}}<br />
== Topological actions ==<br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex>;<br />
Floyd and Richardson \cite{Floyd&Richardson1959} have constructed for the first time a smooth fixed point free action of $G$ on a disk for $G=A_5$, the alternating group on five letters (see {{cite|Bredon1972|pp. 55-58}} for a transparent description of the construction). Next, Greever {{cite|Greever1960}} has described plenty of finite solvable groups $G$, which can act smoothly on disks without fixed points. Then, Oliver \cite{Oliver1975} and \cite{Oliver1976}, has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks. <br />
</wikitex><br />
<br />
==== Oliver number ====<br />
<wikitex>;<br />
Let $G$ be a finite group not of prime power order. Oliver in \cite{Oliver1975} proved that the set $$\{\chi(X^G)-1\colon X\textup{ is a contractible }G\textup{-CW-complex}\}$$ is a subgroup of the group of integers $\mathbb{Z}$. Therefore, the set is of the form $n_G\cdot \mathbb{Z}$ for a unique integer $n_G\geq 0$, which we refer to as the ''Oliver number'' of $G$. <br />
<br />
Oliver has determined integer $n_G$ in the papers \cite{Oliver1975}, \cite{Oliver1977}, and \cite{Oliver1978}. In particular, the following lemma holds.<br />
<br />
{{beginthm|Lemma|(Oliver Lemma)}}<br />
For a finite group $G$ not of prime power order, $n_G=1$ if and only if there does not exist a sequence $P\trianglelefteq H\trianglelefteq G$ of normal subgroups such that $P$ is a $p$-group, $G/H$ is a $q$-group, and $H/P$ is cyclic for two (possibly the same) primes $p$ and $q$.<br />
{{endthm}}<br />
<br />
Moreover, the work {{cite|Oliver1977|Theorem 7}} yields the following proposition.<br />
<br />
{{beginthm|Proposition}}<br />
For a finite nilpotent group $G$ not of prime power order, the following conclusions hold:<br />
<br />
*$n_G=0$ if $G$ has at most one non-cyclic Sylow subgroup.<br />
*$n_G=pq$ for two distinct primes $p$ and $q$, if $G$ has just one non-cyclic $p$-Sylow and $q$-Sylow subgroups.<br />
*$n_G=1$ if $G$ has three or more non-cyclic Sylow subgroups.<br />
<br />
{{endthm}}<br />
<br />
The notion of the Oliver number $n_G$ extends to compact Lie groups $G$ as follows.<br />
<br />
*$n_G=n_{G/G_0}$ if $G_0$ is abelian and $G/G_0$ is not of prime power order.<br />
*$n_G=1$ if $G_0$ is non-abelian (see \cite{Oliver1976}).<br />
<br />
</wikitex><br />
<br />
==== Oliver group ====<br />
<wikitex>;<br />
In connection with the work on [[Group_actions_on_spheres#Smooth_actions|smooth one fixed point actions on spheres]], Laitinen and Morimoto \cite{Laitinen&Morimoto1998} have introduced the notion of Oliver group. <br />
{{beginrem|Definition}}<br />
A finite group $G$ not of prime power order is called an ''Oliver group'' if $n_G=1$.<br />
{{endrem}}<br />
<br />
Examples of finite Oliver groups include:<br />
*$\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ for three distinct primes $p$, $q$, and $r$.<br />
*the groups $S_4\oplus \mathbb{Z}_3$ and $A_4\oplus S_3$ of order 72 (which are solvable but not nilpotent).<br />
*finite non-solvable (in particular, non-trivial perfect) groups (e.g. $A_n$, $S_n$ for $n\geq 5$).<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
The results of Oliver \cite{Oliver1975} and \cite{Oliver1976}, are summarized in the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has a smooth fixed point free action on some disk if and only if at least one of the following condition holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian. <br />
*The quotient $G/G_0$ is not of prime power order and $n_{G/G_0}=1$.<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternions $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ and the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$<br />
$$\widetilde{K}Sp(F)\xleftarrow{q_\mathbb{C}}\widetilde{K}U(F)\xleftarrow{c_\mathbb{R}}\widetilde{K}O(F).$$<br />
<br />
For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$. <br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1975},\cite{Oliver1976})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite CW-complex. Then the following three statements are equivalent.<br />
*The Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$.<br />
*There exist a finite contractible $G$-CW-complex $X$ such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on a disk $D$ such that the fixed point set $D^G$ is homotopy equivalent to $F$.<br />
<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem|\cite{Oliver1996}}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a compact smooth manifold. Then there exists a smooth action of $G$ on some disk $D$ such that the fixed point $D^G$ is diffeomorphic to $F$ if and only if $\chi(F)\equiv 1\pmod{n_G}$ and the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies one of the following condition.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_disksGroup actions on disks2010-11-27T14:12:57Z<p>Marek Kaluba: /* Definitions */</p>
<hr />
<div>{{Stub}}<br />
== Topological actions ==<br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex>;<br />
Floyd and Richardson \cite{Floyd&Richardson1959} have constructed for the first time a smooth fixed point free action of $G$ on a disk for $G=A_5$, the alternating group on five letters (see {{cite|Bredon1972|pp. 55-58}} for a transparent description of the construction). Next, Greever {{cite|Greever1960}} has described plenty of finite solvable groups $G$, which can act smoothly on disks without fixed points. Then, Oliver \cite{Oliver1975} and \cite{Oliver1976}, has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks. <br />
</wikitex><br />
<br />
==== Oliver number ====<br />
<wikitex>;<br />
Let $G$ be a finite group not of prime power order. Oliver in \cite{Oliver1975} proved that the set $$\{\chi(X^G)-1\colon X\textup{ is a contractible }G\textup{-CW-complex}\}$$ is a subgroup of the group of integers $\mathbb{Z}$. Therefore, the set is of the form $n_G\cdot \mathbb{Z}$ for a unique integer $n_G\geq 0$, which we refer to as the ''Oliver number'' of $G$. <br />
<br />
Oliver has determined integer $n_G$ in the papers \cite{Oliver1975}, \cite{Oliver1977}, and \cite{Oliver1978}. In particular, the following lemma holds.<br />
<br />
{{beginthm|Lemma|(Oliver Lemma)}}<br />
For a finite group $G$ not of prime power order, $n_G=1$ if and only if there does not exist a sequence $P\trianglelefteq H\trianglelefteq G$ of normal subgroups such that $P$ is a $p$-group, $G/H$ is a $q$-group, and $H/P$ is cyclic for two (possibly the same) primes $p$ and $q$.<br />
{{endthm}}<br />
<br />
Moreover, the work {{cite|Oliver1977|Theorem 7}} yields the following proposition.<br />
<br />
{{beginthm|Proposition}}<br />
For a finite nilpotent group $G$ not of prime power order, the following conclusions hold:<br />
<br />
*$n_G=0$ if $G$ has at most one non-cyclic Sylow subgroup.<br />
*$n_G=pq$ for two distinct primes $p$ and $q$, if $G$ has just one non-cyclic $p$-Sylow and $q$-Sylow subgroups.<br />
*$n_G=1$ if $G$ has three or more non-cyclic Sylow subgroups.<br />
<br />
{{endthm}}<br />
<br />
The notion of the Oliver number $n_G$ extends to compact Lie groups $G$ as follows.<br />
<br />
*$n_G=n_{G/G_0}$ if $G_0$ is abelian and $G/G_0$ is not of prime power order.<br />
*$n_G=1$ if $G_0$ is non-abelian (see \cite{Oliver1976}).<br />
<br />
</wikitex><br />
<br />
==== Oliver group ====<br />
<wikitex>;<br />
In connection with the work on [[Group_actions_on_spheres#Smooth_actions|smooth one fixed point actions on spheres]], Laitinen and Morimoto \cite{Laitinen&Morimoto1998} have introduced the notion of Oliver group. <br />
{{beginrem|Definition}}<br />
A finite group $G$ not of prime power order is called an ''Oliver group'' if $n_G=1$.<br />
{{endrem}}<br />
<br />
Examples of finite Oliver groups include:<br />
*$\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ for three distinct primes $p$, $q$, and $r$.<br />
*the groups $S_4\oplus \mathbb{Z}_3$ and $A_4\oplus S_3$ of order 72 (which are solvable but not nilpotent).<br />
*finite non-solvable (in particular, non-trivial perfect) groups (e.g. $A_n$, $S_n$ for $n\geq 5$).<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
The results of Oliver \cite{Oliver1975} and \cite{Oliver1976}, are summarized in the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has a smooth fixed point free action on some disk if and only if at least one of the following condition holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian. <br />
*The quotient $G/G_0$ is not of prime power order and $n_{G/G_0}=1$.<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
{{beginrem|Definition}}<br />
<br />
{{endrem}}<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternions $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ and the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$<br />
$$\widetilde{K}Sp(F)\xleftarrow{q_\mathbb{C}}\widetilde{K}U(F)\xleftarrow{c_\mathbb{R}}\widetilde{K}O(F).$$<br />
<br />
Denote by $Tor(A)$ the torsion part of $A$. <br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1975},\cite{Oliver1976})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite CW-complex. Then the following three statements are equivalent.<br />
*The Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$.<br />
*There exist a finite contractible $G$-CW-complex $X$ such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on a disk $D$ such that the fixed point set $D^G$ is homotopy equivalent to $F$.<br />
<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem|\cite{Oliver1996}}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a compact smooth manifold. Then there exists a smooth action of $G$ on some disk $D$ such that the fixed point $D^G$ is diffeomorphic to $F$ if and only if $\chi(F)\equiv 1\pmod{n_G}$ and the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies one of the following condition.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_disksGroup actions on disks2010-11-27T14:09:01Z<p>Marek Kaluba: </p>
<hr />
<div>{{Stub}}<br />
== Topological actions ==<br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex>;<br />
Floyd and Richardson \cite{Floyd&Richardson1959} have constructed for the first time a smooth fixed point free action of $G$ on a disk for $G=A_5$, the alternating group on five letters (see {{cite|Bredon1972|pp. 55-58}} for a transparent description of the construction). Next, Greever {{cite|Greever1960}} has described plenty of finite solvable groups $G$, which can act smoothly on disks without fixed points. Then, Oliver \cite{Oliver1975} and \cite{Oliver1976}, has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks. <br />
</wikitex><br />
<br />
==== Oliver number ====<br />
<wikitex>;<br />
Let $G$ be a finite group not of prime power order. Oliver in \cite{Oliver1975} proved that the set $$\{\chi(X^G)-1\colon X\textup{ is a contractible }G\textup{-CW-complex}\}$$ is a subgroup of the group of integers $\mathbb{Z}$. Therefore, the set is of the form $n_G\cdot \mathbb{Z}$ for a unique integer $n_G\geq 0$, which we refer to as the ''Oliver number'' of $G$. <br />
<br />
Oliver has determined integer $n_G$ in the papers \cite{Oliver1975}, \cite{Oliver1977}, and \cite{Oliver1978}. In particular, the following lemma holds.<br />
<br />
{{beginthm|Lemma|(Oliver Lemma)}}<br />
For a finite group $G$ not of prime power order, $n_G=1$ if and only if there does not exist a sequence $P\trianglelefteq H\trianglelefteq G$ of normal subgroups such that $P$ is a $p$-group, $G/H$ is a $q$-group, and $H/P$ is cyclic for two (possibly the same) primes $p$ and $q$.<br />
{{endthm}}<br />
<br />
Moreover, the work {{cite|Oliver1977|Theorem 7}} yields the following proposition.<br />
<br />
{{beginthm|Proposition}}<br />
For a finite nilpotent group $G$ not of prime power order, the following conclusions hold:<br />
<br />
*$n_G=0$ if $G$ has at most one non-cyclic Sylow subgroup.<br />
*$n_G=pq$ for two distinct primes $p$ and $q$, if $G$ has just one non-cyclic $p$-Sylow and $q$-Sylow subgroups.<br />
*$n_G=1$ if $G$ has three or more non-cyclic Sylow subgroups.<br />
<br />
{{endthm}}<br />
<br />
The notion of the Oliver number $n_G$ extends to compact Lie groups $G$ as follows.<br />
<br />
*$n_G=n_{G/G_0}$ if $G_0$ is abelian and $G/G_0$ is not of prime power order.<br />
*$n_G=1$ if $G_0$ is non-abelian (see \cite{Oliver1976}).<br />
<br />
</wikitex><br />
<br />
==== Oliver group ====<br />
<wikitex>;<br />
In connection with the work on [[Group_actions_on_spheres#Smooth_actions|smooth one fixed point actions on spheres]], Laitinen and Morimoto \cite{Laitinen&Morimoto1998} have introduced the notion of Oliver group. <br />
{{beginrem|Definition}}<br />
A finite group $G$ not of prime power order is called an ''Oliver group'' if $n_G=1$.<br />
{{endrem}}<br />
<br />
Examples of finite Oliver groups include:<br />
*$\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ for three distinct primes $p$, $q$, and $r$.<br />
*the groups $S_4\oplus \mathbb{Z}_3$ and $A_4\oplus S_3$ of order 72 (which are solvable but not nilpotent).<br />
*finite non-solvable (in particular, non-trivial perfect) groups (e.g. $A_n$, $S_n$ for $n\geq 5$).<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
The results of Oliver \cite{Oliver1975} and \cite{Oliver1976}, are summarized in the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has a smooth fixed point free action on some disk if and only if at least one of the following condition holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian. <br />
*The quotient $G/G_0$ is not of prime power order and $n_{G/G_0}=1$.<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
{{beginrem|Definition}}<br />
<br />
{{endrem}}<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternions $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ and the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$<br />
$$\widetilde{K}Sp(F)\xleftarrow{q_\mathbb{C}}\widetilde{K}U(F)\xleftarrow{c_\mathbb{R}}\widetilde{K}O(F).$$<br />
<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1975},\cite{Oliver1976})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite CW-complex. Then the following three statements are equivalent.<br />
*The Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$.<br />
*There exist a finite contractible $G$-CW-complex $X$ such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on a disk $D$ such that the fixed point set $D^G$ is homotopy equivalent to $F$.<br />
<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem|\cite{Oliver1996}}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a compact smooth manifold. Then there exists a smooth action of $G$ on some disk $D$ such that the fixed point $D^G$ is diffeomorphic to $F$ if and only if $\chi(F)\equiv 1\pmod{n_G}$ and the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies one of the following condition.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_disksGroup actions on disks2010-11-27T14:07:29Z<p>Marek Kaluba: /* Definitions */</p>
<hr />
<div>{{Stub}}<br />
== Topological actions ==<br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex>;<br />
Floyd and Richardson \cite{Floyd&Richardson1959} have constructed for the first time a smooth fixed point free action of $G$ on a disk for $G=A_5$, the alternating group on five letters (see {{cite|Bredon1972|pp. 55-58}} for a transparent description of the construction). Next, Greever {{cite|Greever1960}} has described plenty of finite solvable groups $G$, which can act smoothly on disks without fixed points. Then, Oliver \cite{Oliver1975} and \cite{Oliver1976}, has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks. <br />
</wikitex><br />
<br />
==== Oliver number ====<br />
<wikitex>;<br />
Let $G$ be a finite group not of prime power order. Oliver in \cite{Oliver1975} proved that the set $$\{\chi(X^G)-1\colon X\textup{ is a contractible }G\textup{-CW-complex}\}$$ is a subgroup of the group of integers $\mathbb{Z}$. Therefore, the set is of the form $n_G\cdot \mathbb{Z}$ for a unique integer $n_G\geq 0$, which we refer to as the ''Oliver number'' of $G$. <br />
<br />
Oliver has determined integer $n_G$ in the papers \cite{Oliver1975}, \cite{Oliver1977}, and \cite{Oliver1978}. In particular, the following lemma holds.<br />
<br />
{{beginthm|Lemma|(Oliver Lemma)}}<br />
For a finite group $G$ not of prime power order, $n_G=1$ if and only if there does not exist a sequence $P\trianglelefteq H\trianglelefteq G$ of normal subgroups such that $P$ is a $p$-group, $G/H$ is a $q$-group, and $H/P$ is cyclic for two (possibly the same) primes $p$ and $q$.<br />
{{endthm}}<br />
<br />
Moreover, the work {{cite|Oliver1977|Theorem 7}} yields the following proposition.<br />
<br />
{{beginthm|Proposition}}<br />
For a finite nilpotent group $G$ not of prime power order, the following conclusions hold:<br />
<br />
*$n_G=0$ if $G$ has at most one non-cyclic Sylow subgroup.<br />
*$n_G=pq$ for two distinct primes $p$ and $q$, if $G$ has just one non-cyclic $p$-Sylow and $q$-Sylow subgroups.<br />
*$n_G=1$ if $G$ has three or more non-cyclic Sylow subgroups.<br />
<br />
{{endthm}}<br />
<br />
The notion of the Oliver number $n_G$ extends to compact Lie groups $G$ as follows.<br />
<br />
*$n_G=n_{G/G_0}$ if $G_0$ is abelian and $G/G_0$ is not of prime power order.<br />
*$n_G=1$ if $G_0$ is non-abelian (see \cite{Oliver1976}).<br />
<br />
</wikitex><br />
<br />
==== Oliver group ====<br />
<wikitex>;<br />
In connection with the work on [[Group_actions_on_spheres#Smooth_actions|smooth one fixed point actions on spheres]], Laitinen and Morimoto \cite{Laitinen&Morimoto1998} have introduced the notion of Oliver group. <br />
{{beginrem|Definition}}<br />
A finite group $G$ not of prime power order is called an ''Oliver group'' if $n_G=1$.<br />
{{endrem}}<br />
<br />
Examples of finite Oliver groups include:<br />
*$\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ for three distinct primes $p$, $q$, and $r$.<br />
*the groups $S_4\oplus \mathbb{Z}_3$ and $A_4\oplus S_3$ of order 72 (which are solvable but not nilpotent).<br />
*finite non-solvable (in particular, non-trivial perfect) groups (e.g. $A_n$, $S_n$ for $n\geq 5$).<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
The results of Oliver \cite{Oliver1975} and \cite{Oliver1976}, are summarized in the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has a smooth fixed point free action on some disk if and only if at least one of the following condition holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian. <br />
*The quotient $G/G_0$ is not of prime power order and $n_{G/G_0}=1$.<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
{{beginrem|Definition}}<br />
<br />
{{endrem}}<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternions $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ and the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$<br />
$$\widetilde{K}Sp(F)\xleftarrow{q_\mathbb{C}}\widetilde{K}U(F)\xleftarrow{c_\mathbb{R}}\widetilde{K}O(F).$$<br />
<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1975},\cite{Oliver1976})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite CW-complex. Then the following three statements are equivalent.<br />
*The Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$.<br />
*There exist a finite contractible $G$-CW-complex $X$ such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on a disk $D$ such that the fixed point set $D^G$ is homotopy equivalent to $F$.<br />
<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem|\cite{Oliver1996}}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a compact smooth manifold. Then there exists a smooth action of $G$ on some disk $D$ such that the fixed point $D^G$ is diffeomorphic to $F$ if and only if $\chi(F)\equiv 1\pmod{n_G}$ and the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies one of the following condition.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{qDiv}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{qDiv}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kalubahttp://www.map.mpim-bonn.mpg.de/Group_actions_on_disksGroup actions on disks2010-11-27T14:04:55Z<p>Marek Kaluba: /* Results */</p>
<hr />
<div>{{Stub}}<br />
== Topological actions ==<br />
<br />
<br />
== Smooth actions ==<br />
=== Fixed point free ===<br />
==== History ====<br />
<wikitex>;<br />
Floyd and Richardson \cite{Floyd&Richardson1959} have constructed for the first time a smooth fixed point free action of $G$ on a disk for $G=A_5$, the alternating group on five letters (see {{cite|Bredon1972|pp. 55-58}} for a transparent description of the construction). Next, Greever {{cite|Greever1960}} has described plenty of finite solvable groups $G$, which can act smoothly on disks without fixed points. Then, Oliver \cite{Oliver1975} and \cite{Oliver1976}, has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks. <br />
</wikitex><br />
<br />
==== Oliver number ====<br />
<wikitex>;<br />
Let $G$ be a finite group not of prime power order. Oliver in \cite{Oliver1975} proved that the set $$\{\chi(X^G)-1\colon X\textup{ is a contractible }G\textup{-CW-complex}\}$$ is a subgroup of the group of integers $\mathbb{Z}$. Therefore, the set is of the form $n_G\cdot \mathbb{Z}$ for a unique integer $n_G\geq 0$, which we refer to as the ''Oliver number'' of $G$. <br />
<br />
Oliver has determined integer $n_G$ in the papers \cite{Oliver1975}, \cite{Oliver1977}, and \cite{Oliver1978}. In particular, the following lemma holds.<br />
<br />
{{beginthm|Lemma|(Oliver Lemma)}}<br />
For a finite group $G$ not of prime power order, $n_G=1$ if and only if there does not exist a sequence $P\trianglelefteq H\trianglelefteq G$ of normal subgroups such that $P$ is a $p$-group, $G/H$ is a $q$-group, and $H/P$ is cyclic for two (possibly the same) primes $p$ and $q$.<br />
{{endthm}}<br />
<br />
Moreover, the work {{cite|Oliver1977|Theorem 7}} yields the following proposition.<br />
<br />
{{beginthm|Proposition}}<br />
For a finite nilpotent group $G$ not of prime power order, the following conclusions hold:<br />
<br />
*$n_G=0$ if $G$ has at most one non-cyclic Sylow subgroup.<br />
*$n_G=pq$ for two distinct primes $p$ and $q$, if $G$ has just one non-cyclic $p$-Sylow and $q$-Sylow subgroups.<br />
*$n_G=1$ if $G$ has three or more non-cyclic Sylow subgroups.<br />
<br />
{{endthm}}<br />
<br />
The notion of the Oliver number $n_G$ extends to compact Lie groups $G$ as follows.<br />
<br />
*$n_G=n_{G/G_0}$ if $G_0$ is abelian and $G/G_0$ is not of prime power order.<br />
*$n_G=1$ if $G_0$ is non-abelian (see \cite{Oliver1976}).<br />
<br />
</wikitex><br />
<br />
==== Oliver group ====<br />
<wikitex>;<br />
In connection with the work on [[Group_actions_on_spheres#Smooth_actions|smooth one fixed point actions on spheres]], Laitinen and Morimoto \cite{Laitinen&Morimoto1998} have introduced the notion of Oliver group. <br />
{{beginrem|Definition}}<br />
A finite group $G$ not of prime power order is called an ''Oliver group'' if $n_G=1$.<br />
{{endrem}}<br />
<br />
Examples of finite Oliver groups include:<br />
*$\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ for three distinct primes $p$, $q$, and $r$.<br />
*the groups $S_4\oplus \mathbb{Z}_3$ and $A_4\oplus S_3$ of order 72 (which are solvable but not nilpotent).<br />
*finite non-solvable (in particular, non-trivial perfect) groups (e.g. $A_n$, $S_n$ for $n\geq 5$).<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
The results of Oliver \cite{Oliver1975} and \cite{Oliver1976}, are summarized in the following theorem.<br />
<br />
{{beginthm|Theorem}}<br />
A compact Lie group $G$ has a smooth fixed point free action on some disk if and only if at least one of the following condition holds.<br />
*The identity connected component $G_0$ of $G$ is non-abelian. <br />
*The quotient $G/G_0$ is not of prime power order and $n_{G/G_0}=1$.<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
=== Fixed point sets ===<br />
==== History ====<br />
<br />
==== Definitions ====<br />
<wikitex>;<br />
Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.<br />
<br />
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.<br />
<br />
*$\mathcal{A}=\{G\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.<br />
*$\mathcal{B}=\{G\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.<br />
*$\mathcal{C}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
*$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.<br />
*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.<br />
<br />
{{beginrm|Definition}}<br />
<br />
{{endrm}}<br />
<br />
Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternions $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:<br />
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ and the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$<br />
$$\widetilde{K}Sp(F)\xleftarrow{q_\mathbb{C}}\widetilde{K}U(F)\xleftarrow{c_\mathbb{R}}\widetilde{K}O(F).$$<br />
<br />
</wikitex><br />
<br />
==== Results ====<br />
<wikitex>;<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.<br />
<br />
{{beginthm|Theorem|(\cite{Oliver1975},\cite{Oliver1976})}}<br />
Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite CW-complex. Then the following three statements are equivalent.<br />
*The Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$.<br />
*There exist a finite contractible $G$-CW-complex $X$ such that the fixed point set $X^G$ is homeomorphic to $F$.<br />
*There exists a smooth action of $G$ on a disk $D$ such that the fixed point set $D^G$ is homotopy equivalent to $F$.<br />
<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem|\cite{Oliver1996}}}<br />
Let $G$ be a finite group not of prime power order. Let $F$ be a compact smooth manifold. Then there exists a smooth action of $G$ on some disk $D$ such that the fixed point $D^G$ is diffeomorphic to $F$ if and only if $\chi(F)\equiv 1\pmod{n_G}$ and the class $[\tau_F]\in \widetilde{K}O(F)$ satisfies one of the following condition.<br />
<br />
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.<br />
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{qDiv}(\widetilde{K}U(F))$.<br />
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.<br />
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{qDiv}(\widetilde{K}O(F))$.<br />
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{qDiv}(\widetilde{K}U(F)))$.<br />
<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]<br />
[[Category:Group actions on manifolds]]</div>Marek Kaluba