5-manifolds with fundamental group of order 2

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[edit] 1 Introduction

The classification of simply-connected 5-manifolds was achieved by Smale [Smale] and Barden [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 ([Markov], [Kervaire]) that every finitely generated group can be realized as the fundamental group of a manifold of dimension $ {{Stub}} == Introduction == <wikitex>; ... </wikitex> == Construction and examples == <wikitex>; ... </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex>  == References == {{#RefList:}}   [[Category:Manifolds]] [[Catefory:5-Manifolds]]\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.

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.)

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.

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 [Hambleton&Su] .

[edit] 2 Construction and examples

First some examples known from other context.

$S^2 \times \mathbb R P^3$ ;
$X^5(q)$ $q=1,3,5,7$ 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".

[edit] 2.1 Circle bundles over simply-connected 4-manifolds

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

[Hambleton&Su, Proposition 4.2] 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 $0$ .

$k=1$ , $M$ 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 [Duan&Liang].
$k=2$ , 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 [Hambleton&Su].

[edit] 2.2 Connected sum along $S^1$

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.

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 $M_1 \sharp_{S^1} M_2$ .

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.

[edit] 3 Invariants

...

[edit] 4 Classification/Characterization

...

[edit] 5 Further discussion

...

[edit] 6 References

5-manifolds with fundamental group of order 2

Latest revision as of 02:36, 7 February 2013

Contents

[edit] 1 Introduction

[edit] 2 Construction and examples

[edit] 2.1 Circle bundles over simply-connected 4-manifolds

[edit] 2.2 Connected sum along $S^1$

[edit] 3 Invariants

[edit] 4 Classification/Characterization

[edit] 5 Further discussion

[edit] 6 References

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@@ Line 1: / Line 1: @@
+{{Stub}}
+== Introduction ==
+<wikitex>;
+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.
-<!-- COMMENT:
+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.)
-To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:
+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.
-- For statements like Theorem, Lemma, Definition etc., use e.g.
+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} .
-{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.
-- For references, use e.g. {{cite|Milnor1958b}}.
+</wikitex>
-END OF COMMENT
+== Construction and examples ==
+<wikitex>;
+First some examples known from other context.
+*$S^2 \times \mathbb R P^3$;
+* $X^5(q)$ $q=1,3,5,7$ 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".
+</wikitex>
--->
-{{Stub}}
+=== Circle bundles over simply-connected 4-manifolds ===
-== Introduction ==
 <wikitex>;
-...
+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
+{{beginthm|Proposition}}\cite[Proposition 4.2]{Hambleton&Su}
+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 $0$.
+{{endthm}}
+* $k=1$, $M$ 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}.
+* $k=2$, 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}.
 </wikitex>
-== Construction and examples ==
+=== Connected sum along $S^1$ ===
 <wikitex>;
-...
+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.
+{{beginthm|Definition}}
+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 $M_1 \sharp_{S^1} M_2$.
+{{endthm}}
+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.
 </wikitex>