http://www.map.mpim-bonn.mpg.de/api.php?action=feedcontributions&user=Martin+Olbermann&feedformat=atomManifold Atlas - User contributions [en]2022-11-29T10:05:16ZUser contributionsMediaWiki 1.18.4http://www.map.mpim-bonn.mpg.de/String_bordismString bordism2013-02-11T16:03:03Z<p>Martin Olbermann: /* Singular homology */</p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
$String$-bordism or $O\!\left< 8 \right>$-bordism is a special case of a [[B-Bordism|B-bordism]]. It comes from the tower of fibrations below.<br />
$$<br />
\xymatrix{BO\!\left< 8 \right>\ar[d] & \\ BSpin \ar[d]\ar[r]^{p_1/2}&K({\mathbb Z},4) \\ BSO\ar[d]\ar[r]^{w_2}& K({\mathbb Z}/2,2) \\ BO\ar[r]^{w_1}&<br />
K({\mathbb Z}/2,1) }<br />
$$<br />
In each step the lowest homotopy group is killed by the map into the Eilenberg-MacLane spaces. In particular, $BO\left< 8 \right>$ is the homotopy fibre of the map from $BSpin$ given by half of the first Pontryagin class. The name $String$-group is due to Haynes Miller and will be explained below.<br />
<br />
</wikitex><br />
<br />
== The String group ==<br />
<wikitex>;<br />
There are various models for the String group. However, since its first three homotopy groups vanish it can not be realized as a finite dimensional Lie group. There are homotopy theoretic constructions which involve path spaces. A more geometric way to think about $String(n)$ which involves only finite dimensional manifolds is the following: the String group fibers over the Spin group with fibre $K({\mathbb Z},2)$. One may think of $ K({\mathbb Z},2)$ as the realization of $S^1$ viewed as a smooth category with only one object. This way, the $A_\infty$ space $String(n)$ appears as the realization of a smooth 2-group extension of $Spin(n)$ by the finite dimensional Lie groupoid $S^1$ (see \cite{Schommer-Pries2009}). A more explicit model for this extension can be found in \cite{Meinrenken2003}. <br />
</wikitex><br />
<br />
== The bordism groups ==<br />
<wikitex>;<br />
The additive structure of the bordism groups is not fully determined yet. It is known that only 2 and 3 torsion appears (see \cite{Giambalvo1971}) and the 3 torsion is annihalated by multiplication with 3 (see \cite{Hovey1997}). Moreover, the bordism groups $\Omega_{k}^{String}$ are finite for $k=1,2,3$ mod 4.<br />
<br />
Clearly, since $BO\!\left< 8 \right>$ is 7-connected the first 6 bordism groups coincide with the [[Framed bordism|framed bordism groups]]. The first 16 bordism groups have been computed by Giambalvo \cite{Giambalvo1971|p. 538}:<br />
* $\Omega_7^{String} = 0$.<br />
* $\Omega_8^{String} \cong \Zz \oplus \Zz/2$, generated by the [[Exotic spheres|exotic 8-sphere]] for the 2-torsion and a certain [[Bott manifold]]: see \cite{Laures2004}.<br />
* $\Omega_9^{String} \cong \Theta_9/bP_{10} \cong (\Zz/2)^2$, generated by [[Exotic spheres|exotic 9-spheres]].<br />
* $\Omega_{10}^{String} \cong \Zz/6$, generated by an [[Exotic spheres|exotic 10-sphere]].<br />
* $\Omega_{11}^{String} = 0$.<br />
* $\Omega_{12}^{String} \cong \Zz$, generated by a 5-connected manifold with signature $8 \times 992$.<br />
* $\Omega_{13}^{String} \cong \Zz/3$, generated by an [[Exotic spheres|exotic 13-sphere]].<br />
* $\Omega_{14}^{String} \cong \Zz/2$, generated by the [[Exotic spheres|exotic 14-sphere]].<br />
* $\Omega_{15}^{String} \cong \Zz/2$, genreated by the [[Exotic spheres|exotic 15-sphere]].<br />
* $\Omega_{16}^{String} \cong \Zz^2$.<br />
At the prime 3 the first 32 bordism groups can be found in \cite{Hovey&Ravenel1995}. Further calculations have been done in \cite{Mahowald&Gorbounov1995}. <br />
</wikitex><br />
<br />
== Homology calculations ==<br />
===Singular homology===<br />
<wikitex>;<br />
The cohomology ring $H^*(BString,{\mathbb Z}/p)$ has been computed for $p=2$ by Stong in \cite{Stong1963}: <br />
$$ H^*(BString)\cong {\mathbb Z}/2[\theta_i\mid \sigma_2(i-1)\ge 4]\otimes H^*(K({\mathbb Z},8))/Sq^2(\iota).$$<br />
Here, $\sigma_2$ is the number of ones in the duadic decomposition and the $\theta_i$ come from the cohomology of $BO$ and coincide with the Stiefel-Whitney up to decomposables. <br />
<br />
From Stong's proof it follows that $ H^*(BO)\to H^*(BString)$ is surjective, and $H^*(BString)\cong {\mathbb Z}/2[\theta_i\mid \sigma_2(i-1)\ge 3]$ is a polynomial algebra. <br />
<br />
For odd $p$ the corresponding result has been obtained by Giambalvo \cite{Giambalvo1969}.<br />
<br />
</wikitex><br />
<br />
===K(1)-local computations===<br />
<wikitex>;<br />
$K(1)$ locally $MString$ coincides with $MSpin$ and decomposes into a wedge of copies of $KO$. However, it is not an algebra over $KO$. Its multiplicative structure for $p=2$ can be read off the formula<br />
$$ L_{K(1)}MString \cong T_\zeta \wedge \bigwedge TS^0.$$<br />
Here, $\zeta\in \pi_{-1}L_{K(1)}S^0$ is a generator, $T_\zeta$ is the $E_\infty $ cone over $\zeta$ and $TS^0$ is the free $E_\infty$ spectrum generated by the sphere. In particular, its $\theta$-algebra structure is free (see \cite{Laures2003a}).<br />
<br />
</wikitex><br />
<br />
===K(n)-homology computations===<br />
<br />
<wikitex>;<br />
For Morava $K=K(n)$ at $p=2$ one has an exact sequence of Hopf algebras (see \cite{Kitchloo&Laures&Wilson2004a})<br />
$$ \xymatrix{K_* \ar[r] & K_*K({\mathbb Z}/2,2)\ar[r]& K_*K({\mathbb Z},3)\ar[r]& K_*BString \ar[r]&K_*BSpin \ar[r]& K_* K({\mathbb Z},4)\ar[r]^2& K({\mathbb Z},4)\ar[r]&K_*}$$<br />
which is induced by the obvious geometric maps. For $n=2$ it algebraically reduces to the split exact sequence of Hopf algebras (see \cite{Kitchloo&Laures&Wilson2004b})<br />
$$\xymatrix{K(2)_* \ar[r] & K(2)_*K({\mathbb Z},3)\ar[r]& K(2)_*BString \ar[r]&K(2)_*BSpin \ar[r]& K(2)_*}.$$<br />
</wikitex><br />
<br />
===Computations with respect to general complex oriented theories===<br />
<wikitex>;<br />
Ando, Hopkins and Strickland investigated the homology ring $E_*BString$ for even periodic multiplicative cohomology theories $E$. Even periodic theories are complex orientable which means that $E^0({\mathbb C}P^\infty)$ carries a formal group. The description of $E_*BString$ is in terms of formal group data. <br />
<br />
In \cite{Ando&Hopkins&Strickland2001a} first the analogous complex problem is studied. The group $O\left< 8 \right>=String$ has a complex relative $U\left< 6 \right>$ which is defined in the same way by killing the third homotopy group of $SU$. Consider the map<br />
$$ (1-L_1)(1-L_2)(1-L_3):<br />
\xymatrix{ {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty\ar[r]& BU}$$<br />
where the $L_i$ are the canonical line bundles over the individual factors. Since the first two Chern classes of the virtual bundle vanish the map lifts to $BU\left<6 \right>$. If we choose a complex orienatation the lift gives a class $f$ in the cohomology <br />
ring<br />
$$(E\wedge BU\left<6 \right>_+)^0( {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty )\cong <br />
E_0BU\left<6 \right> [\![x_1,x_2,x_3]\!]$$ <br />
with $c_i(L_i)=x_i$. The power series $f$ satisfies the following identities:<br />
$$ \begin{aligned}<br />
f(0,0,0)&=1\\<br />
f(x_1,x_2,x_3)&=f(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)});\; \sigma \in \Sigma_3\\<br />
f(x_1,x_2,x_3)f(x_0,x_1+_Fx_2,x_3)&=f(x_0+_Fx_1,x_2,x_3)f(x_0,x_1,x_2).<br />
\end{aligned}$$<br />
Hence, the power series is an example a symmetric 3-cocycle or a cubical structure on the formal group law. In fact the main result of \cite{Ando&Hopkins&Strickland2001a} is that it is the universal example of such a structure. Explicitly, this means that the commutative ring $E_0 BU\left <6 \right>$ is freely generated by the coefficients of $f$ subject to the relations given by the 3 equations above.<br />
<br />
The real version of this result has not been published yet by the three authors. Using the diagram<br />
$$ \xymatrix{K{\mathbb Z},3)\ar[r]\ar[d]^=&BU\left< 6 \right>\ar[r]\ar[d]& BSU\ar[d]\\<br />
K({\mathbb Z},3)\ar[r]& BString \ar[r]&BSpin}$$<br />
and the results for $K(2)_*BString$ described above they conjecture that $E_0BString$ is the same quotient subject to the additional relation<br />
$$ f(x_1,x_2,x_3)=1 \mbox{ for } x_1+_Fx_2+_Fx_3=0.$$<br />
</wikitex><br />
<br />
== The structure of the spectrum==<br />
<wikitex>;<br />
Localized at a prime $p>3$, string bordism splits additively into a sum of suspensions of $BP$, although the ring structure is different (see \cite{Hovey2008}). For $p=3$ there is a spectrum $Y$ with 3 cells in even dimensions such that $MString\wedge Y$ splits into a sum of suspensions of $BP$. For $p=2$ it is hoped that the spectrum $tmf$ splits off which is explained below. <br />
<br />
</wikitex><br />
<br />
== The Witten genus ==<br />
<wikitex>;<br />
At the end of the 80s Ed Witten were studying the $S^1$-equivariant index of the Dirac operator on a loop space of a $4k$-dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space $LM$ this would mean that $M$ is $String$. Witten carried the Atiyah-Segal formula for the index over to this infinite dimensional setting and obtained an integral modular form of weight $k$. Nowadays this is called the Witten genus (see \cite{Segal1988}.)<br />
The Witten genus can be refined to a map of structured ring spectra<br />
$$W: MString \longrightarrow TMF$$<br />
from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms (\cite{Hopkins2002}). This map is also called the $\sigma$-orientation and is 15-connected (see \cite{Hill2008}). The spectrum $TMF$ was developed by Goerss, Hopkins and Miller. It is supposed to play the same role for $String$-bordism as $KO$-theory does for $Spin$-bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map $W$ gives characteristic numbers which together with $KO$ and Stiefel-Whitney numbers are conjectured to determine the $String$ bordism class. Moreover, $tmf$ is supposed to be a direct summand of $MString$ as the orientation map $W$ is shown to be surjective in homotopy (see \cite{Hopkins&Mahowald2002}.)<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]<br />
[[Category:Bordism]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/String_bordismString bordism2013-02-11T12:22:11Z<p>Martin Olbermann: /* Singular homology */</p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
$String$-bordism or $O\!\left< 8 \right>$-bordism is a special case of a [[B-Bordism|B-bordism]]. It comes from the tower of fibrations below.<br />
$$<br />
\xymatrix{BO\!\left< 8 \right>\ar[d] & \\ BSpin \ar[d]\ar[r]^{p_1/2}&K({\mathbb Z},4) \\ BSO\ar[d]\ar[r]^{w_2}& K({\mathbb Z}/2,2) \\ BO\ar[r]^{w_1}&<br />
K({\mathbb Z}/2,1) }<br />
$$<br />
In each step the lowest homotopy group is killed by the map into the Eilenberg-MacLane spaces. In particular, $BO\left< 8 \right>$ is the homotopy fibre of the map from $BSpin$ given by half of the first Pontryagin class. The name $String$-group is due to Haynes Miller and will be explained below.<br />
<br />
</wikitex><br />
<br />
== The String group ==<br />
<wikitex>;<br />
There are various models for the String group. However, since its first three homotopy groups vanish it can not be realized as a finite dimensional Lie group. There are homotopy theoretic constructions which involve path spaces. A more geometric way to think about $String(n)$ which involves only finite dimensional manifolds is the following: the String group fibers over the Spin group with fibre $K({\mathbb Z},2)$. One may think of $ K({\mathbb Z},2)$ as the realization of $S^1$ viewed as a smooth category with only one object. This way, the $A_\infty$ space $String(n)$ appears as the realization of a smooth 2-group extension of $Spin(n)$ by the finite dimensional Lie groupoid $S^1$ (see \cite{Schommer-Pries2009}). A more explicit model for this extension can be found in \cite{Meinrenken2003}. <br />
</wikitex><br />
<br />
== The bordism groups ==<br />
<wikitex>;<br />
The additive structure of the bordism groups is not fully determined yet. It is known that only 2 and 3 torsion appears (see \cite{Giambalvo1971}) and the 3 torsion is annihalated by multiplication with 3 (see \cite{Hovey1997}). Moreover, the bordism groups $\Omega_{k}^{String}$ are finite for $k=1,2,3$ mod 4.<br />
<br />
Clearly, since $BO\!\left< 8 \right>$ is 7-connected the first 6 bordism groups coincide with the [[Framed bordism|framed bordism groups]]. The first 16 bordism groups have been computed by Giambalvo \cite{Giambalvo1971|p. 538}:<br />
* $\Omega_7^{String} = 0$.<br />
* $\Omega_8^{String} \cong \Zz \oplus \Zz/2$, generated by the [[Exotic spheres|exotic 8-sphere]] for the 2-torsion and a certain [[Bott manifold]]: see \cite{Laures2004}.<br />
* $\Omega_9^{String} \cong \Theta_9/bP_{10} \cong (\Zz/2)^2$, generated by [[Exotic spheres|exotic 9-spheres]].<br />
* $\Omega_{10}^{String} \cong \Zz/6$, generated by an [[Exotic spheres|exotic 10-sphere]].<br />
* $\Omega_{11}^{String} = 0$.<br />
* $\Omega_{12}^{String} \cong \Zz$, generated by a 5-connected manifold with signature $8 \times 992$.<br />
* $\Omega_{13}^{String} \cong \Zz/3$, generated by an [[Exotic spheres|exotic 13-sphere]].<br />
* $\Omega_{14}^{String} \cong \Zz/2$, generated by the [[Exotic spheres|exotic 14-sphere]].<br />
* $\Omega_{15}^{String} \cong \Zz/2$, genreated by the [[Exotic spheres|exotic 15-sphere]].<br />
* $\Omega_{16}^{String} \cong \Zz^2$.<br />
At the prime 3 the first 32 bordism groups can be found in \cite{Hovey&Ravenel1995}. Further calculations have been done in \cite{Mahowald&Gorbounov1995}. <br />
</wikitex><br />
<br />
== Homology calculations ==<br />
===Singular homology===<br />
<wikitex>;<br />
The cohomology ring $H^*(BString,{\mathbb Z}/p)$ has been computed for $p=2$ by Stong in \cite{Stong1963}: <br />
$$ H^*(BString)\cong {\mathbb Z}/2[\theta_i\mid \sigma_2(i-1)\ge 4]\otimes H^*(K({\mathbb Z},8))/Sq^2(\iota).$$<br />
Here, $\sigma_2$ is the number of ones in the duadic decomposition and the $\theta_i$ come from the cohomology of $BO$ and coincide with the Stiefel-Whitney up to decomposables. For odd $p$ the corresponding result has been obtained by Giambalvo \cite{Giambalvo1969}.<br />
<br />
</wikitex><br />
<br />
===K(1)-local computations===<br />
<wikitex>;<br />
$K(1)$ locally $MString$ coincides with $MSpin$ and decomposes into a wedge of copies of $KO$. However, it is not an algebra over $KO$. Its multiplicative structure for $p=2$ can be read off the formula<br />
$$ L_{K(1)}MString \cong T_\zeta \wedge \bigwedge TS^0.$$<br />
Here, $\zeta\in \pi_{-1}L_{K(1)}S^0$ is a generator, $T_\zeta$ is the $E_\infty $ cone over $\zeta$ and $TS^0$ is the free $E_\infty$ spectrum generated by the sphere. In particular, its $\theta$-algebra structure is free (see \cite{Laures2003a}).<br />
<br />
</wikitex><br />
<br />
===K(n)-homology computations===<br />
<br />
<wikitex>;<br />
For Morava $K=K(n)$ at $p=2$ one has an exact sequence of Hopf algebras (see \cite{Kitchloo&Laures&Wilson2004a})<br />
$$ \xymatrix{K_* \ar[r] & K_*K({\mathbb Z}/2,2)\ar[r]& K_*K({\mathbb Z},3)\ar[r]& K_*BString \ar[r]&K_*BSpin \ar[r]& K_* K({\mathbb Z},4)\ar[r]^2& K({\mathbb Z},4)\ar[r]&K_*}$$<br />
which is induced by the obvious geometric maps. For $n=2$ it algebraically reduces to the split exact sequence of Hopf algebras (see \cite{Kitchloo&Laures&Wilson2004b})<br />
$$\xymatrix{K(2)_* \ar[r] & K(2)_*K({\mathbb Z},3)\ar[r]& K(2)_*BString \ar[r]&K(2)_*BSpin \ar[r]& K(2)_*}.$$<br />
</wikitex><br />
<br />
===Computations with respect to general complex oriented theories===<br />
<wikitex>;<br />
Ando, Hopkins and Strickland investigated the homology ring $E_*BString$ for even periodic multiplicative cohomology theories $E$. Even periodic theories are complex orientable which means that $E^0({\mathbb C}P^\infty)$ carries a formal group. The description of $E_*BString$ is in terms of formal group data. <br />
<br />
In \cite{Ando&Hopkins&Strickland2001a} first the analogous complex problem is studied. The group $O\left< 8 \right>=String$ has a complex relative $U\left< 6 \right>$ which is defined in the same way by killing the third homotopy group of $SU$. Consider the map<br />
$$ (1-L_1)(1-L_2)(1-L_3):<br />
\xymatrix{ {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty\ar[r]& BU}$$<br />
where the $L_i$ are the canonical line bundles over the individual factors. Since the first two Chern classes of the virtual bundle vanish the map lifts to $BU\left<6 \right>$. If we choose a complex orienatation the lift gives a class $f$ in the cohomology <br />
ring<br />
$$(E\wedge BU\left<6 \right>_+)^0( {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty )\cong <br />
E_0BU\left<6 \right> [\![x_1,x_2,x_3]\!]$$ <br />
with $c_i(L_i)=x_i$. The power series $f$ satisfies the following identities:<br />
$$ \begin{aligned}<br />
f(0,0,0)&=1\\<br />
f(x_1,x_2,x_3)&=f(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)});\; \sigma \in \Sigma_3\\<br />
f(x_1,x_2,x_3)f(x_0,x_1+_Fx_2,x_3)&=f(x_0+_Fx_1,x_2,x_3)f(x_0,x_1,x_2).<br />
\end{aligned}$$<br />
Hence, the power series is an example a symmetric 3-cocycle or a cubical structure on the formal group law. In fact the main result of \cite{Ando&Hopkins&Strickland2001a} is that it is the universal example of such a structure. Explicitly, this means that the commutative ring $E_0 BU\left <6 \right>$ is freely generated by the coefficients of $f$ subject to the relations given by the 3 equations above.<br />
<br />
The real version of this result has not been published yet by the three authors. Using the diagram<br />
$$ \xymatrix{K{\mathbb Z},3)\ar[r]\ar[d]^=&BU\left< 6 \right>\ar[r]\ar[d]& BSU\ar[d]\\<br />
K({\mathbb Z},3)\ar[r]& BString \ar[r]&BSpin}$$<br />
and the results for $K(2)_*BString$ described above they conjecture that $E_0BString$ is the same quotient subject to the additional relation<br />
$$ f(x_1,x_2,x_3)=1 \mbox{ for } x_1+_Fx_2+_Fx_3=0.$$<br />
</wikitex><br />
<br />
== The structure of the spectrum==<br />
<wikitex>;<br />
Localized at a prime $p>3$, string bordism splits additively into a sum of suspensions of $BP$, although the ring structure is different (see \cite{Hovey2008}). For $p=3$ there is a spectrum $Y$ with 3 cells in even dimensions such that $MString\wedge Y$ splits into a sum of suspensions of $BP$. For $p=2$ it is hoped that the spectrum $tmf$ splits off which is explained below. <br />
<br />
</wikitex><br />
<br />
== The Witten genus ==<br />
<wikitex>;<br />
At the end of the 80s Ed Witten were studying the $S^1$-equivariant index of the Dirac operator on a loop space of a $4k$-dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space $LM$ this would mean that $M$ is $String$. Witten carried the Atiyah-Segal formula for the index over to this infinite dimensional setting and obtained an integral modular form of weight $k$. Nowadays this is called the Witten genus (see \cite{Segal1988}.)<br />
The Witten genus can be refined to a map of structured ring spectra<br />
$$W: MString \longrightarrow TMF$$<br />
from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms (\cite{Hopkins2002}). This map is also called the $\sigma$-orientation and is 15-connected (see \cite{Hill2008}). The spectrum $TMF$ was developed by Goerss, Hopkins and Miller. It is supposed to play the same role for $String$-bordism as $KO$-theory does for $Spin$-bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map $W$ gives characteristic numbers which together with $KO$ and Stiefel-Whitney numbers are conjectured to determine the $String$ bordism class. Moreover, $tmf$ is supposed to be a direct summand of $MString$ as the orientation map $W$ is shown to be surjective in homotopy (see \cite{Hopkins&Mahowald2002}.)<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]<br />
[[Category:Bordism]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/String_bordismString bordism2013-02-11T12:21:07Z<p>Martin Olbermann: /* Singular homology */</p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
$String$-bordism or $O\!\left< 8 \right>$-bordism is a special case of a [[B-Bordism|B-bordism]]. It comes from the tower of fibrations below.<br />
$$<br />
\xymatrix{BO\!\left< 8 \right>\ar[d] & \\ BSpin \ar[d]\ar[r]^{p_1/2}&K({\mathbb Z},4) \\ BSO\ar[d]\ar[r]^{w_2}& K({\mathbb Z}/2,2) \\ BO\ar[r]^{w_1}&<br />
K({\mathbb Z}/2,1) }<br />
$$<br />
In each step the lowest homotopy group is killed by the map into the Eilenberg-MacLane spaces. In particular, $BO\left< 8 \right>$ is the homotopy fibre of the map from $BSpin$ given by half of the first Pontryagin class. The name $String$-group is due to Haynes Miller and will be explained below.<br />
<br />
</wikitex><br />
<br />
== The String group ==<br />
<wikitex>;<br />
There are various models for the String group. However, since its first three homotopy groups vanish it can not be realized as a finite dimensional Lie group. There are homotopy theoretic constructions which involve path spaces. A more geometric way to think about $String(n)$ which involves only finite dimensional manifolds is the following: the String group fibers over the Spin group with fibre $K({\mathbb Z},2)$. One may think of $ K({\mathbb Z},2)$ as the realization of $S^1$ viewed as a smooth category with only one object. This way, the $A_\infty$ space $String(n)$ appears as the realization of a smooth 2-group extension of $Spin(n)$ by the finite dimensional Lie groupoid $S^1$ (see \cite{Schommer-Pries2009}). A more explicit model for this extension can be found in \cite{Meinrenken2003}. <br />
</wikitex><br />
<br />
== The bordism groups ==<br />
<wikitex>;<br />
The additive structure of the bordism groups is not fully determined yet. It is known that only 2 and 3 torsion appears (see \cite{Giambalvo1971}) and the 3 torsion is annihalated by multiplication with 3 (see \cite{Hovey1997}). Moreover, the bordism groups $\Omega_{k}^{String}$ are finite for $k=1,2,3$ mod 4.<br />
<br />
Clearly, since $BO\!\left< 8 \right>$ is 7-connected the first 6 bordism groups coincide with the [[Framed bordism|framed bordism groups]]. The first 16 bordism groups have been computed by Giambalvo \cite{Giambalvo1971|p. 538}:<br />
* $\Omega_7^{String} = 0$.<br />
* $\Omega_8^{String} \cong \Zz \oplus \Zz/2$, generated by the [[Exotic spheres|exotic 8-sphere]] for the 2-torsion and a certain [[Bott manifold]]: see \cite{Laures2004}.<br />
* $\Omega_9^{String} \cong \Theta_9/bP_{10} \cong (\Zz/2)^2$, generated by [[Exotic spheres|exotic 9-spheres]].<br />
* $\Omega_{10}^{String} \cong \Zz/6$, generated by an [[Exotic spheres|exotic 10-sphere]].<br />
* $\Omega_{11}^{String} = 0$.<br />
* $\Omega_{12}^{String} \cong \Zz$, generated by a 5-connected manifold with signature $8 \times 992$.<br />
* $\Omega_{13}^{String} \cong \Zz/3$, generated by an [[Exotic spheres|exotic 13-sphere]].<br />
* $\Omega_{14}^{String} \cong \Zz/2$, generated by the [[Exotic spheres|exotic 14-sphere]].<br />
* $\Omega_{15}^{String} \cong \Zz/2$, genreated by the [[Exotic spheres|exotic 15-sphere]].<br />
* $\Omega_{16}^{String} \cong \Zz^2$.<br />
At the prime 3 the first 32 bordism groups can be found in \cite{Hovey&Ravenel1995}. Further calculations have been done in \cite{Mahowald&Gorbounov1995}. <br />
</wikitex><br />
<br />
== Homology calculations ==<br />
===Singular homology===<br />
<wikitex>;<br />
The cohomology ring $H^*(BString,{\mathbb Z}/p)$ has been computed for $p=2$ by Stong in \cite{Stong1963}: <br />
$$ H^*(BString)\cong {\mathbb Z}/2[\theta_i\mid \sigma_2(i-1)\ge 4]\otimes H^*(K({\mathbb Z},4))/Sq^2(\iota).$$<br />
Here, $\sigma_2$ is the number of ones in the duadic decomposition and the $\theta_i$ come from the cohomology of $BO$ and coincide with the Stiefel-Whitney up to decomposables. For odd $p$ the corresponding result has been obtained by Giambalvo \cite{Giambalvo1969}.<br />
<br />
</wikitex><br />
<br />
===K(1)-local computations===<br />
<wikitex>;<br />
$K(1)$ locally $MString$ coincides with $MSpin$ and decomposes into a wedge of copies of $KO$. However, it is not an algebra over $KO$. Its multiplicative structure for $p=2$ can be read off the formula<br />
$$ L_{K(1)}MString \cong T_\zeta \wedge \bigwedge TS^0.$$<br />
Here, $\zeta\in \pi_{-1}L_{K(1)}S^0$ is a generator, $T_\zeta$ is the $E_\infty $ cone over $\zeta$ and $TS^0$ is the free $E_\infty$ spectrum generated by the sphere. In particular, its $\theta$-algebra structure is free (see \cite{Laures2003a}).<br />
<br />
</wikitex><br />
<br />
===K(n)-homology computations===<br />
<br />
<wikitex>;<br />
For Morava $K=K(n)$ at $p=2$ one has an exact sequence of Hopf algebras (see \cite{Kitchloo&Laures&Wilson2004a})<br />
$$ \xymatrix{K_* \ar[r] & K_*K({\mathbb Z}/2,2)\ar[r]& K_*K({\mathbb Z},3)\ar[r]& K_*BString \ar[r]&K_*BSpin \ar[r]& K_* K({\mathbb Z},4)\ar[r]^2& K({\mathbb Z},4)\ar[r]&K_*}$$<br />
which is induced by the obvious geometric maps. For $n=2$ it algebraically reduces to the split exact sequence of Hopf algebras (see \cite{Kitchloo&Laures&Wilson2004b})<br />
$$\xymatrix{K(2)_* \ar[r] & K(2)_*K({\mathbb Z},3)\ar[r]& K(2)_*BString \ar[r]&K(2)_*BSpin \ar[r]& K(2)_*}.$$<br />
</wikitex><br />
<br />
===Computations with respect to general complex oriented theories===<br />
<wikitex>;<br />
Ando, Hopkins and Strickland investigated the homology ring $E_*BString$ for even periodic multiplicative cohomology theories $E$. Even periodic theories are complex orientable which means that $E^0({\mathbb C}P^\infty)$ carries a formal group. The description of $E_*BString$ is in terms of formal group data. <br />
<br />
In \cite{Ando&Hopkins&Strickland2001a} first the analogous complex problem is studied. The group $O\left< 8 \right>=String$ has a complex relative $U\left< 6 \right>$ which is defined in the same way by killing the third homotopy group of $SU$. Consider the map<br />
$$ (1-L_1)(1-L_2)(1-L_3):<br />
\xymatrix{ {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty\ar[r]& BU}$$<br />
where the $L_i$ are the canonical line bundles over the individual factors. Since the first two Chern classes of the virtual bundle vanish the map lifts to $BU\left<6 \right>$. If we choose a complex orienatation the lift gives a class $f$ in the cohomology <br />
ring<br />
$$(E\wedge BU\left<6 \right>_+)^0( {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty )\cong <br />
E_0BU\left<6 \right> [\![x_1,x_2,x_3]\!]$$ <br />
with $c_i(L_i)=x_i$. The power series $f$ satisfies the following identities:<br />
$$ \begin{aligned}<br />
f(0,0,0)&=1\\<br />
f(x_1,x_2,x_3)&=f(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)});\; \sigma \in \Sigma_3\\<br />
f(x_1,x_2,x_3)f(x_0,x_1+_Fx_2,x_3)&=f(x_0+_Fx_1,x_2,x_3)f(x_0,x_1,x_2).<br />
\end{aligned}$$<br />
Hence, the power series is an example a symmetric 3-cocycle or a cubical structure on the formal group law. In fact the main result of \cite{Ando&Hopkins&Strickland2001a} is that it is the universal example of such a structure. Explicitly, this means that the commutative ring $E_0 BU\left <6 \right>$ is freely generated by the coefficients of $f$ subject to the relations given by the 3 equations above.<br />
<br />
The real version of this result has not been published yet by the three authors. Using the diagram<br />
$$ \xymatrix{K{\mathbb Z},3)\ar[r]\ar[d]^=&BU\left< 6 \right>\ar[r]\ar[d]& BSU\ar[d]\\<br />
K({\mathbb Z},3)\ar[r]& BString \ar[r]&BSpin}$$<br />
and the results for $K(2)_*BString$ described above they conjecture that $E_0BString$ is the same quotient subject to the additional relation<br />
$$ f(x_1,x_2,x_3)=1 \mbox{ for } x_1+_Fx_2+_Fx_3=0.$$<br />
</wikitex><br />
<br />
== The structure of the spectrum==<br />
<wikitex>;<br />
Localized at a prime $p>3$, string bordism splits additively into a sum of suspensions of $BP$, although the ring structure is different (see \cite{Hovey2008}). For $p=3$ there is a spectrum $Y$ with 3 cells in even dimensions such that $MString\wedge Y$ splits into a sum of suspensions of $BP$. For $p=2$ it is hoped that the spectrum $tmf$ splits off which is explained below. <br />
<br />
</wikitex><br />
<br />
== The Witten genus ==<br />
<wikitex>;<br />
At the end of the 80s Ed Witten were studying the $S^1$-equivariant index of the Dirac operator on a loop space of a $4k$-dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space $LM$ this would mean that $M$ is $String$. Witten carried the Atiyah-Segal formula for the index over to this infinite dimensional setting and obtained an integral modular form of weight $k$. Nowadays this is called the Witten genus (see \cite{Segal1988}.)<br />
The Witten genus can be refined to a map of structured ring spectra<br />
$$W: MString \longrightarrow TMF$$<br />
from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms (\cite{Hopkins2002}). This map is also called the $\sigma$-orientation and is 15-connected (see \cite{Hill2008}). The spectrum $TMF$ was developed by Goerss, Hopkins and Miller. It is supposed to play the same role for $String$-bordism as $KO$-theory does for $Spin$-bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map $W$ gives characteristic numbers which together with $KO$ and Stiefel-Whitney numbers are conjectured to determine the $String$ bordism class. Moreover, $tmf$ is supposed to be a direct summand of $MString$ as the orientation map $W$ is shown to be surjective in homotopy (see \cite{Hopkins&Mahowald2002}.)<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]<br />
[[Category:Bordism]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/String_bordismString bordism2013-02-11T12:18:53Z<p>Martin Olbermann: /* Singular homology */</p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
$String$-bordism or $O\!\left< 8 \right>$-bordism is a special case of a [[B-Bordism|B-bordism]]. It comes from the tower of fibrations below.<br />
$$<br />
\xymatrix{BO\!\left< 8 \right>\ar[d] & \\ BSpin \ar[d]\ar[r]^{p_1/2}&K({\mathbb Z},4) \\ BSO\ar[d]\ar[r]^{w_2}& K({\mathbb Z}/2,2) \\ BO\ar[r]^{w_1}&<br />
K({\mathbb Z}/2,1) }<br />
$$<br />
In each step the lowest homotopy group is killed by the map into the Eilenberg-MacLane spaces. In particular, $BO\left< 8 \right>$ is the homotopy fibre of the map from $BSpin$ given by half of the first Pontryagin class. The name $String$-group is due to Haynes Miller and will be explained below.<br />
<br />
</wikitex><br />
<br />
== The String group ==<br />
<wikitex>;<br />
There are various models for the String group. However, since its first three homotopy groups vanish it can not be realized as a finite dimensional Lie group. There are homotopy theoretic constructions which involve path spaces. A more geometric way to think about $String(n)$ which involves only finite dimensional manifolds is the following: the String group fibers over the Spin group with fibre $K({\mathbb Z},2)$. One may think of $ K({\mathbb Z},2)$ as the realization of $S^1$ viewed as a smooth category with only one object. This way, the $A_\infty$ space $String(n)$ appears as the realization of a smooth 2-group extension of $Spin(n)$ by the finite dimensional Lie groupoid $S^1$ (see \cite{Schommer-Pries2009}). A more explicit model for this extension can be found in \cite{Meinrenken2003}. <br />
</wikitex><br />
<br />
== The bordism groups ==<br />
<wikitex>;<br />
The additive structure of the bordism groups is not fully determined yet. It is known that only 2 and 3 torsion appears (see \cite{Giambalvo1971}) and the 3 torsion is annihalated by multiplication with 3 (see \cite{Hovey1997}). Moreover, the bordism groups $\Omega_{k}^{String}$ are finite for $k=1,2,3$ mod 4.<br />
<br />
Clearly, since $BO\!\left< 8 \right>$ is 7-connected the first 6 bordism groups coincide with the [[Framed bordism|framed bordism groups]]. The first 16 bordism groups have been computed by Giambalvo \cite{Giambalvo1971|p. 538}:<br />
* $\Omega_7^{String} = 0$.<br />
* $\Omega_8^{String} \cong \Zz \oplus \Zz/2$, generated by the [[Exotic spheres|exotic 8-sphere]] for the 2-torsion and a certain [[Bott manifold]]: see \cite{Laures2004}.<br />
* $\Omega_9^{String} \cong \Theta_9/bP_{10} \cong (\Zz/2)^2$, generated by [[Exotic spheres|exotic 9-spheres]].<br />
* $\Omega_{10}^{String} \cong \Zz/6$, generated by an [[Exotic spheres|exotic 10-sphere]].<br />
* $\Omega_{11}^{String} = 0$.<br />
* $\Omega_{12}^{String} \cong \Zz$, generated by a 5-connected manifold with signature $8 \times 992$.<br />
* $\Omega_{13}^{String} \cong \Zz/3$, generated by an [[Exotic spheres|exotic 13-sphere]].<br />
* $\Omega_{14}^{String} \cong \Zz/2$, generated by the [[Exotic spheres|exotic 14-sphere]].<br />
* $\Omega_{15}^{String} \cong \Zz/2$, genreated by the [[Exotic spheres|exotic 15-sphere]].<br />
* $\Omega_{16}^{String} \cong \Zz^2$.<br />
At the prime 3 the first 32 bordism groups can be found in \cite{Hovey&Ravenel1995}. Further calculations have been done in \cite{Mahowald&Gorbounov1995}. <br />
</wikitex><br />
<br />
== Homology calculations ==<br />
===Singular homology===<br />
<wikitex>;<br />
The cohomology ring $H^*(BString,{\mathbb Z}/p)$ has been computed for $p=2$ by Stong in \cite{Stong1963}: <br />
$$ H^*(BString)\cong {\mathbb Z}/2[\theta_i\mid \sigma_2(i-1)>4]\otimes H^*(K({\mathbb Z},4))/Sq^2(\iota).$$<br />
Here, $\sigma_2$ is the number of ones in the duadic decomposition and the $\theta_i$ come from the cohomology of $BO$ and coincide with the Stiefel-Whitney up to decomposables. For odd $p$ the corresponding result has been obtained by Giambalvo \cite{Giambalvo1969}.<br />
<br />
</wikitex><br />
<br />
===K(1)-local computations===<br />
<wikitex>;<br />
$K(1)$ locally $MString$ coincides with $MSpin$ and decomposes into a wedge of copies of $KO$. However, it is not an algebra over $KO$. Its multiplicative structure for $p=2$ can be read off the formula<br />
$$ L_{K(1)}MString \cong T_\zeta \wedge \bigwedge TS^0.$$<br />
Here, $\zeta\in \pi_{-1}L_{K(1)}S^0$ is a generator, $T_\zeta$ is the $E_\infty $ cone over $\zeta$ and $TS^0$ is the free $E_\infty$ spectrum generated by the sphere. In particular, its $\theta$-algebra structure is free (see \cite{Laures2003a}).<br />
<br />
</wikitex><br />
<br />
===K(n)-homology computations===<br />
<br />
<wikitex>;<br />
For Morava $K=K(n)$ at $p=2$ one has an exact sequence of Hopf algebras (see \cite{Kitchloo&Laures&Wilson2004a})<br />
$$ \xymatrix{K_* \ar[r] & K_*K({\mathbb Z}/2,2)\ar[r]& K_*K({\mathbb Z},3)\ar[r]& K_*BString \ar[r]&K_*BSpin \ar[r]& K_* K({\mathbb Z},4)\ar[r]^2& K({\mathbb Z},4)\ar[r]&K_*}$$<br />
which is induced by the obvious geometric maps. For $n=2$ it algebraically reduces to the split exact sequence of Hopf algebras (see \cite{Kitchloo&Laures&Wilson2004b})<br />
$$\xymatrix{K(2)_* \ar[r] & K(2)_*K({\mathbb Z},3)\ar[r]& K(2)_*BString \ar[r]&K(2)_*BSpin \ar[r]& K(2)_*}.$$<br />
</wikitex><br />
<br />
===Computations with respect to general complex oriented theories===<br />
<wikitex>;<br />
Ando, Hopkins and Strickland investigated the homology ring $E_*BString$ for even periodic multiplicative cohomology theories $E$. Even periodic theories are complex orientable which means that $E^0({\mathbb C}P^\infty)$ carries a formal group. The description of $E_*BString$ is in terms of formal group data. <br />
<br />
In \cite{Ando&Hopkins&Strickland2001a} first the analogous complex problem is studied. The group $O\left< 8 \right>=String$ has a complex relative $U\left< 6 \right>$ which is defined in the same way by killing the third homotopy group of $SU$. Consider the map<br />
$$ (1-L_1)(1-L_2)(1-L_3):<br />
\xymatrix{ {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty\ar[r]& BU}$$<br />
where the $L_i$ are the canonical line bundles over the individual factors. Since the first two Chern classes of the virtual bundle vanish the map lifts to $BU\left<6 \right>$. If we choose a complex orienatation the lift gives a class $f$ in the cohomology <br />
ring<br />
$$(E\wedge BU\left<6 \right>_+)^0( {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty )\cong <br />
E_0BU\left<6 \right> [\![x_1,x_2,x_3]\!]$$ <br />
with $c_i(L_i)=x_i$. The power series $f$ satisfies the following identities:<br />
$$ \begin{aligned}<br />
f(0,0,0)&=1\\<br />
f(x_1,x_2,x_3)&=f(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)});\; \sigma \in \Sigma_3\\<br />
f(x_1,x_2,x_3)f(x_0,x_1+_Fx_2,x_3)&=f(x_0+_Fx_1,x_2,x_3)f(x_0,x_1,x_2).<br />
\end{aligned}$$<br />
Hence, the power series is an example a symmetric 3-cocycle or a cubical structure on the formal group law. In fact the main result of \cite{Ando&Hopkins&Strickland2001a} is that it is the universal example of such a structure. Explicitly, this means that the commutative ring $E_0 BU\left <6 \right>$ is freely generated by the coefficients of $f$ subject to the relations given by the 3 equations above.<br />
<br />
The real version of this result has not been published yet by the three authors. Using the diagram<br />
$$ \xymatrix{K{\mathbb Z},3)\ar[r]\ar[d]^=&BU\left< 6 \right>\ar[r]\ar[d]& BSU\ar[d]\\<br />
K({\mathbb Z},3)\ar[r]& BString \ar[r]&BSpin}$$<br />
and the results for $K(2)_*BString$ described above they conjecture that $E_0BString$ is the same quotient subject to the additional relation<br />
$$ f(x_1,x_2,x_3)=1 \mbox{ for } x_1+_Fx_2+_Fx_3=0.$$<br />
</wikitex><br />
<br />
== The structure of the spectrum==<br />
<wikitex>;<br />
Localized at a prime $p>3$, string bordism splits additively into a sum of suspensions of $BP$, although the ring structure is different (see \cite{Hovey2008}). For $p=3$ there is a spectrum $Y$ with 3 cells in even dimensions such that $MString\wedge Y$ splits into a sum of suspensions of $BP$. For $p=2$ it is hoped that the spectrum $tmf$ splits off which is explained below. <br />
<br />
</wikitex><br />
<br />
== The Witten genus ==<br />
<wikitex>;<br />
At the end of the 80s Ed Witten were studying the $S^1$-equivariant index of the Dirac operator on a loop space of a $4k$-dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space $LM$ this would mean that $M$ is $String$. Witten carried the Atiyah-Segal formula for the index over to this infinite dimensional setting and obtained an integral modular form of weight $k$. Nowadays this is called the Witten genus (see \cite{Segal1988}.)<br />
The Witten genus can be refined to a map of structured ring spectra<br />
$$W: MString \longrightarrow TMF$$<br />
from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms (\cite{Hopkins2002}). This map is also called the $\sigma$-orientation and is 15-connected (see \cite{Hill2008}). The spectrum $TMF$ was developed by Goerss, Hopkins and Miller. It is supposed to play the same role for $String$-bordism as $KO$-theory does for $Spin$-bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map $W$ gives characteristic numbers which together with $KO$ and Stiefel-Whitney numbers are conjectured to determine the $String$ bordism class. Moreover, $tmf$ is supposed to be a direct summand of $MString$ as the orientation map $W$ is shown to be surjective in homotopy (see \cite{Hopkins&Mahowald2002}.)<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]<br />
[[Category:Bordism]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/String_bordismString bordism2013-02-07T12:15:17Z<p>Martin Olbermann: /* Singular homology */</p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
$String$-bordism or $O\!\left< 8 \right>$-bordism is a special case of a [[B-Bordism|B-bordism]]. It comes from the tower of fibrations below.<br />
$$<br />
\xymatrix{BO\!\left< 8 \right>\ar[d] & \\ BSpin \ar[d]\ar[r]^{p_1/2}&K({\mathbb Z},4) \\ BSO\ar[d]\ar[r]^{w_2}& K({\mathbb Z}/2,2) \\ BO\ar[r]^{w_1}&<br />
K({\mathbb Z}/2,1) }<br />
$$<br />
In each step the lowest homotopy group is killed by the map into the Eilenberg-MacLane spaces. In particular, $BO\left< 8 \right>$ is the homotopy fibre of the map from $BSpin$ given by half of the first Pontryagin class. The name $String$-group is due to Haynes Miller and will be explained below.<br />
<br />
</wikitex><br />
<br />
== The String group ==<br />
<wikitex>;<br />
There are various models for the String group. However, since its first three homotopy groups vanish it can not be realized as a finite dimensional Lie group. There are homotopy theoretic constructions which involve path spaces. A more geometric way to think about $String(n)$ which involves only finite dimensional manifolds is the following: the String group fibers over the Spin group with fibre $K({\mathbb Z},2)$. One may think of $ K({\mathbb Z},2)$ as the realization of $S^1$ viewed as a smooth category with only one object. This way, the $A_\infty$ space $String(n)$ appears as the realization of a smooth 2-group extension of $Spin(n)$ by the finite dimensional Lie groupoid $S^1$ (see \cite{Schommer-Pries2009}). A more explicit model for this extension can be found in \cite{Meinrenken2003}. <br />
</wikitex><br />
<br />
== The bordism groups ==<br />
<wikitex>;<br />
The additive structure of the bordism groups is not fully determined yet. It is known that only 2 and 3 torsion appears (see \cite{Giambalvo1971}) and the 3 torsion is annihalated by multiplication with 3 (see \cite{Hovey1997}). Moreover, the bordism groups $\Omega_{k}^{String}$ are finite for $k=1,2,3$ mod 4.<br />
<br />
Clearly, since $BO\!\left< 8 \right>$ is 7-connected the first 6 bordism groups coincide with the [[Framed bordism|framed bordism groups]]. The first 16 bordism groups have been computed by Giambalvo \cite{Giambalvo1971|p. 538}:<br />
* $\Omega_7^{String} = 0$.<br />
* $\Omega_8^{String} \cong \Zz \oplus \Zz/2$, generated by the [[Exotic spheres|exotic 8-sphere]] for the 2-torsion and a certain [[Bott manifold]]: see \cite{Laures2004}.<br />
* $\Omega_9^{String} \cong \Theta_9/bP_{10} \cong (\Zz/2)^2$, generated by [[Exotic spheres|exotic 9-spheres]].<br />
* $\Omega_{10}^{String} \cong \Zz/6$, generated by an [[Exotic spheres|exotic 10-sphere]].<br />
* $\Omega_{11}^{String} = 0$.<br />
* $\Omega_{12}^{String} \cong \Zz$, generated by a 5-connected manifold with signature $8 \times 992$.<br />
* $\Omega_{13}^{String} \cong \Zz/3$, generated by an [[Exotic spheres|exotic 13-sphere]].<br />
* $\Omega_{14}^{String} \cong \Zz/2$, generated by the [[Exotic spheres|exotic 14-sphere]].<br />
* $\Omega_{15}^{String} \cong \Zz/2$, genreated by the [[Exotic spheres|exotic 15-sphere]].<br />
* $\Omega_{16}^{String} \cong \Zz^2$.<br />
At the prime 3 the first 32 bordism groups can be found in \cite{Hovey&Ravenel1995}. Further calculations have been done in \cite{Mahowald&Gorbounov1995}. <br />
</wikitex><br />
<br />
== Homology calculations ==<br />
===Singular homology===<br />
<wikitex>;<br />
The cohomology ring $H^*(BString,{\mathbb Z}/p)$ has been computed for $p=2$ by Stong in \cite{Stong1963}: <br />
$$ H^*(BString)\cong {\mathbb Z}/2[\theta_i|\sigma_2(i-1)>4]\otimes H^*(K({\mathbb Z},4))/Sq^2(\iota).$$<br />
Here, $\sigma_2$ is the number of ones in the duadic decomposition and the $\theta_i$ come from the cohomology of $BO$ and coincide with the Stiefel-Whitney up to decomposables. For odd $p$ the corresponding result has been obtained by Giambalvo \cite{Giambalvo1969}.<br />
<br />
</wikitex><br />
<br />
===K(1)-local computations===<br />
<wikitex>;<br />
$K(1)$ locally $MString$ coincides with $MSpin$ and decomposes into a wedge of copies of $KO$. However, it is not an algebra over $KO$. Its multiplicative structure for $p=2$ can be read off the formula<br />
$$ L_{K(1)}MString \cong T_\zeta \wedge \bigwedge TS^0.$$<br />
Here, $\zeta\in \pi_{-1}L_{K(1)}S^0$ is a generator, $T_\zeta$ is the $E_\infty $ cone over $\zeta$ and $TS^0$ is the free $E_\infty$ spectrum generated by the sphere. In particular, its $\theta$-algebra structure is free (see \cite{Laures2003a}).<br />
<br />
</wikitex><br />
<br />
===K(n)-homology computations===<br />
<br />
<wikitex>;<br />
For Morava $K=K(n)$ at $p=2$ one has an exact sequence of Hopf algebras (see \cite{Kitchloo&Laures&Wilson2004a})<br />
$$ \xymatrix{K_* \ar[r] & K_*K({\mathbb Z}/2,2)\ar[r]& K_*K({\mathbb Z},3)\ar[r]& K_*BString \ar[r]&K_*BSpin \ar[r]& K_* K({\mathbb Z},4)\ar[r]^2& K({\mathbb Z},4)\ar[r]&K_*}$$<br />
which is induced by the obvious geometric maps. For $n=2$ it algebraically reduces to the split exact sequence of Hopf algebras (see \cite{Kitchloo&Laures&Wilson2004b})<br />
$$\xymatrix{K(2)_* \ar[r] & K(2)_*K({\mathbb Z},3)\ar[r]& K(2)_*BString \ar[r]&K(2)_*BSpin \ar[r]& K(2)_*}.$$<br />
</wikitex><br />
<br />
===Computations with respect to general complex oriented theories===<br />
<wikitex>;<br />
Ando, Hopkins and Strickland investigated the homology ring $E_*BString$ for even periodic multiplicative cohomology theories $E$. Even periodic theories are complex orientable which means that $E^0({\mathbb C}P^\infty)$ carries a formal group. The description of $E_*BString$ is in terms of formal group data. <br />
<br />
In \cite{Ando&Hopkins&Strickland2001a} first the analogous complex problem is studied. The group $O\left< 8 \right>=String$ has a complex relative $U\left< 6 \right>$ which is defined in the same way by killing the third homotopy group of $SU$. Consider the map<br />
$$ (1-L_1)(1-L_2)(1-L_3):<br />
\xymatrix{ {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty\ar[r]& BU}$$<br />
where the $L_i$ are the canonical line bundles over the individual factors. Since the first two Chern classes of the virtual bundle vanish the map lifts to $BU\left<6 \right>$. If we choose a complex orienatation the lift gives a class $f$ in the cohomology <br />
ring<br />
$$(E\wedge BU\left<6 \right>_+)^0( {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty )\cong <br />
E_0BU\left<6 \right> [\![x_1,x_2,x_3]\!]$$ <br />
with $c_i(L_i)=x_i$. The power series $f$ satisfies the following identities:<br />
$$ \begin{aligned}<br />
f(0,0,0)&=1\\<br />
f(x_1,x_2,x_3)&=f(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)});\; \sigma \in \Sigma_3\\<br />
f(x_1,x_2,x_3)f(x_0,x_1+_Fx_2,x_3)&=f(x_0+_Fx_1,x_2,x_3)f(x_0,x_1,x_2).<br />
\end{aligned}$$<br />
Hence, the power series is an example a symmetric 3-cocycle or a cubical structure on the formal group law. In fact the main result of \cite{Ando&Hopkins&Strickland2001a} is that it is the universal example of such a structure. Explicitly, this means that the commutative ring $E_0 BU\left <6 \right>$ is freely generated by the coefficients of $f$ subject to the relations given by the 3 equations above.<br />
<br />
The real version of this result has not been published yet by the three authors. Using the diagram<br />
$$ \xymatrix{K{\mathbb Z},3)\ar[r]\ar[d]^=&BU\left< 6 \right>\ar[r]\ar[d]& BSU\ar[d]\\<br />
K({\mathbb Z},3)\ar[r]& BString \ar[r]&BSpin}$$<br />
and the results for $K(2)_*BString$ described above they conjecture that $E_0BString$ is the same quotient subject to the additional relation<br />
$$ f(x_1,x_2,x_3)=1 \mbox{ for } x_1+_Fx_2+_Fx_3=0.$$<br />
</wikitex><br />
<br />
== The structure of the spectrum==<br />
<wikitex>;<br />
Localized at a prime $p>3$, string bordism splits additively into a sum of suspensions of $BP$, although the ring structure is different (see \cite{Hovey2008}). For $p=3$ there is a spectrum $Y$ with 3 cells in even dimensions such that $MString\wedge Y$ splits into a sum of suspensions of $BP$. For $p=2$ it is hoped that the spectrum $tmf$ splits off which is explained below. <br />
<br />
</wikitex><br />
<br />
== The Witten genus ==<br />
<wikitex>;<br />
At the end of the 80s Ed Witten were studying the $S^1$-equivariant index of the Dirac operator on a loop space of a $4k$-dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space $LM$ this would mean that $M$ is $String$. Witten carried the Atiyah-Segal formula for the index over to this infinite dimensional setting and obtained an integral modular form of weight $k$. Nowadays this is called the Witten genus (see \cite{Segal1988}.)<br />
The Witten genus can be refined to a map of structured ring spectra<br />
$$W: MString \longrightarrow TMF$$<br />
from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms (\cite{Hopkins2002}). This map is also called the $\sigma$-orientation and is 15-connected (see \cite{Hill2008}). The spectrum $TMF$ was developed by Goerss, Hopkins and Miller. It is supposed to play the same role for $String$-bordism as $KO$-theory does for $Spin$-bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map $W$ gives characteristic numbers which together with $KO$ and Stiefel-Whitney numbers are conjectured to determine the $String$ bordism class. Moreover, $tmf$ is supposed to be a direct summand of $MString$ as the orientation map $W$ is shown to be surjective in homotopy (see \cite{Hopkins&Mahowald2002}.)<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]<br />
[[Category:Bordism]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Chain_duality_III_(Ex)Chain duality III (Ex)2012-06-01T15:59:57Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
Let $F : \Aa \rightarrow \Aa'$ be a functor of additive categories with chain duality. Show that the assignment<br />
$$<br />
M \otimes_{\Aa} N \rightarrow F(M) \otimes_{\Aa'} F(N) <br />
$$<br />
given by<br />
$$<br />
\varphi \colon TM \rightarrow N \quad \mapsto \quad F (\varphi) \circ G(M) \colon T' F (M) \rightarrow FT(M) \rightarrow F(N)<br />
$$<br />
induces a $\Zz_2$-equivariant chain map <br />
$$<br />
C \otimes_{\Aa} C \rightarrow F(C) \otimes_{\Aa'} F(C)<br />
$$<br />
for any $C \in \Bb(\Aa)$.<br />
</wikitex><br />
== References ==<br />
{{#RefList:}}<br />
[[Category:Exercises]]<br />
[[Category:Exercises without solution]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Chain_duality_III_(Ex)Talk:Chain duality III (Ex)2012-06-01T10:35:20Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
The most interesting part is to check equivariance, say for objects $M\in\mathbb A$.<br />
<br />
Let $\varphi:TM\to M$ be an element of $M\otimes_{\mathbb A}M$.<br />
<br />
We have to check the equality of<br />
$$T'_{F(M),F(M)}(F (\varphi) \circ G(M))=e'_{F(M)}\circ T'G(M) \circ T'F(\varphi)$$<br />
and $$F(T_{M,M}\varphi)\circ G(M)=F(e_M)\circ FT\varphi\circ G(M).$$<br />
<br />
This follows from the commutative diagram<br />
$$\xymatrix{<br />
T'F(M)\ar[r]^{T'F(\varphi)} \ar[d]_{G(M)} & <br />
T'FT(M) \ar[r]^{T'G(M)} \ar[d]_{G(TM)} &<br />
T'^2F(M) \ar[d]_{e'_{F(M)}}\\<br />
FT(M)\ar[r]^{FT\varphi} &<br />
FT^2M\ar[r]^{Fe_M}&<br />
F(M)<br />
}$$<br />
as the first square commutes by naturality of $G$ and the second one by definition of a<br />
functor of categories with chain duality.<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Chain_duality_III_(Ex)Talk:Chain duality III (Ex)2012-06-01T10:34:01Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We check this for objects $M\in\mathbb A$.<br />
<br />
Let $\varphi:TM\to M$ be an element of $M\otimes_{\mathbb A}M$.<br />
<br />
We have to check the equality of<br />
$$T'_{F(M),F(M)}(F (\varphi) \circ G(M))=e'_{F(M)}\circ T'G(M) \circ T'F(\varphi)$$<br />
and $$F(T_{M,M}\varphi)\circ G(M)=F(e_M)\circ FT\varphi\circ G(M).$$<br />
<br />
This follows from the commutative diagram<br />
$$\xymatrix{<br />
T'F(M)\ar[r]^{T'F(\varphi)} \ar[d]_{G(M)} & <br />
T'FT(M) \ar[r]^{T'G(M)} \ar[d]_{G(TM)} &<br />
T'^2F(M) \ar[d]_{e'_{F(M)}}\\<br />
FT(M)\ar[r]^{FT\varphi} &<br />
FT^2M\ar[r]^{Fe_M}&<br />
F(M)<br />
}$$<br />
as the first square commutes by naturality of $G$ and the second one by definition of a<br />
functor of categories with chain duality.<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Chain_duality_III_(Ex)Talk:Chain duality III (Ex)2012-06-01T10:32:39Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We check this for objects $M\in\mathbb A$.<br />
<br />
Let $\varphi:TM\to M$ be an element of $M\otimes_{\mathbb A}M$.<br />
<br />
We have to check the equality of<br />
$$T'_{F(M),F(M)}(F (\varphi) \circ G(M))=e'_{F(M)}\circ T'G(M) \circ T'F(\varphi)$$<br />
and $$F(T_{M,M}\varphi)\circ G(M)=F(e_M)\circ FT\varphi\circ G(M).$$<br />
<br />
This follows from the commutative diagram<br />
$$\xymatrix{<br />
T'F(M)\ar[r]^{T'F(\varphi)} \ar[d]_{G(M)} & <br />
T'FT(M) \ar[r]^{T'G(M)} \ar[d]_{G(TM)} &<br />
T'^2F(M) \ar[d]_{e'_{F(M)}}\\<br />
FT(M)\ar[r]^{FT\varphi} &<br />
FT^2M\ar[r]^{Fe_M}&<br />
F(M)<br />
}.$$<br />
<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Chain_duality_III_(Ex)Talk:Chain duality III (Ex)2012-06-01T10:22:20Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We check this for objects $M\in\mathbb A$.<br />
<br />
Let $\varphi:TM\to M$ be an element of $M\otimes_{\mathbb A}M$.<br />
<br />
We have to check the equality of<br />
$T'_{F(M),F(M)}(F (\varphi) \circ G(M))=e'_{F(M)}\circ T'G(M) \circ T'F(\varphi)$<br />
and $F(T_{M,M}\varphi)\circ G(M)=F(e_M)\circ FT\varphi\circ G(M)$.<br />
<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Chain_duality_III_(Ex)Talk:Chain duality III (Ex)2012-06-01T10:21:50Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We check this for objects $M\in\mathbb A$.<br />
<br />
Let $\varphi:TM\to M$ be an element of $M\otimes_{\mathbb A}M$.<br />
%Then $T_{M,M}\varphi=e_M\circ T\varphi: TM\to T^2M\to M$.<br />
<br />
We have to check the equality of<br />
$T'_{F(M),F(M)}(F (\varphi) \circ G(M))=e'_{F(M)}\circ T'G(M) \circ T'F(\varphi)$<br />
and $F(T_{M,M}\varphi)\circ G(M)=F(e_M)\circ FT\varphi\circ G(M)$.<br />
<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Chain_duality_III_(Ex)Talk:Chain duality III (Ex)2012-06-01T10:14:00Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We check this for objects in $\mathbb A$.<br />
<br />
Let $\phi:TM\to M$ be an element of $M\otimes_{\mathbb A}M$.<br />
Then $T_{M,M}\phi=(e_M\circ T\phi: TM\to T^2M\to M$.<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Chain_duality_III_(Ex)Talk:Chain duality III (Ex)2012-06-01T10:13:32Z<p>Martin Olbermann: Created page with "We check this for objects in $\mathbb A$. Let $\phi:TM\to M$ be an element of $M\otimes_{\mathbb A}M$. Then $T_{M,M}\phi=(e_M\circ T\phi: TM\to T^2M\to M$."</p>
<hr />
<div>We check this for objects in $\mathbb A$.<br />
<br />
Let $\phi:TM\to M$ be an element of $M\otimes_{\mathbb A}M$.<br />
Then $T_{M,M}\phi=(e_M\circ T\phi: TM\to T^2M\to M$.</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Oberwolfach_Surgery_Seminar_2012:_ExercisesOberwolfach Surgery Seminar 2012: Exercises2012-06-01T10:10:56Z<p>Martin Olbermann: /* Talks 11-13 */</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:Martin Olbermann|Martin Olbermann]]<br />
# [[Normal bundles in products of spheres (Ex)]]<br />
# [[Non-reducible Spivak Normal Fibrations (Ex)]] - [[User:Steve Balady|Steve Balady]]<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)]] - [[User: Rovi|Carmen Rovi]]<br />
<br />
# [[Surgery obstruction map I (Ex)]] - [[User:Martin Olbermann|Martin Olbermann]]<br />
# [[Structures on M x I (Ex)]] - [[User:Daniel Kasprowski|Daniel Kasprowski]]<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)]] - [[User:Spiros Adams-Florou|Spiros Adams-Florou]]<br />
# [[Chain duality II (Ex)]] - [[User:Markullmann|Mark Ullmann]]<br />
# [[Chain duality III (Ex)]] - [[User:Martin Olbermann|Martin Olbermann]]<br />
# [[Chain duality IV (Ex)]]<br />
# [[Chain duality V (Ex)]]<br />
# [[Chain duality VI (Ex)]]<br />
# [[Chain duality VII (Ex)]]<br />
# [[Supplement I (Ex)]] - [[User:Martin Palmer|Martin Palmer]]<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>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Chain_duality_III_(Ex)Chain duality III (Ex)2012-06-01T10:05:34Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
Let $F \colon \Aa \rightarrow \Aa'$ be a functor of additive categories with chain duality. Show that the assignment<br />
$$<br />
M \otimes_{\Aa} N \rightarrow F(M) \otimes_{\Aa'} F(N) <br />
$$<br />
given by<br />
$$<br />
\varphi \colon TM \rightarrow N \quad \mapsto \quad F (\varphi) \circ G(M) \colon T' F (M) \rightarrow FT(M) \rightarrow F(N)<br />
$$<br />
induces a $\Zz_2$-equivariant chain map <br />
$$<br />
C \otimes_{\Aa} C \rightarrow F(C) \otimes_{\Aa'} F(C)<br />
$$<br />
for any $C \in \Bb(\Aa)$.<br />
</wikitex><br />
== References ==<br />
{{#RefList:}}<br />
[[Category:Exercises]]<br />
[[Category:Exercises without solution]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Chain_duality_V_(Ex)Chain duality V (Ex)2012-05-31T16:08:48Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
Let $\Aa$ be an additive category with chain duality $(T,e)$. Show that under the isomorphism<br />
$$<br />
T_{M,TM} \colon M \otimes_{\Aa} TM \rightarrow TM \otimes_{\Aa} M<br />
$$<br />
we have<br />
$$<br />
T_{M,TM} (\textup{id}) = e_{M}.<br />
$$<br />
Observe that $e$ can be defined this way if we have already defined $T$ and $T_{M,N}$.<br />
</wikitex><br />
== References ==<br />
{{#RefList:}}<br />
[[Category:Exercises]]<br />
[[Category:Exercises with solution]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Chain_duality_V_(Ex)Talk:Chain duality V (Ex)2012-05-31T16:07:37Z<p>Martin Olbermann: Created page with "<wikitex>; This is immediate from the definition that $T_{M,N}(\phi:TM\to N)=e_M\circ T(\phi)$. </wikitex>"</p>
<hr />
<div><wikitex>;<br />
This is immediate from the definition that $T_{M,N}(\phi:TM\to N)=e_M\circ T(\phi)$.<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Oberwolfach_Surgery_Seminar_2012:_ExercisesOberwolfach Surgery Seminar 2012: Exercises2012-05-31T16:05:48Z<p>Martin Olbermann: /* Talks 11-13 */</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:Martin Olbermann|Martin Olbermann]]<br />
# [[Normal bundles in products of spheres (Ex)]]<br />
# [[Non-reducible Spivak Normal Fibrations (Ex)]] - [[User:Steve Balady|Steve Balady]]<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)]] - - [[User:Martin Olbermann|Martin Olbermann]]<br />
# [[Structures on M x I (Ex)]] - [[User:Daniel Kasprowski|Daniel Kasprowski]]<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)]] - [[User:Spiros Adams-Florou|Spiros Adams-Florou]]<br />
# [[Chain duality II (Ex)]]<br />
# [[Chain duality III (Ex)]]<br />
# [[Chain duality IV (Ex)]]<br />
# [[Chain duality V (Ex)]]- [[User:Martin Olbermann|Martin Olbermann]]<br />
# [[Chain duality VI (Ex)]]<br />
# [[Chain duality VII (Ex)]]<br />
# [[Supplement I (Ex)]] - [[User:Martin Palmer|Martin Palmer]]<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>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Oberwolfach_Surgery_Seminar_2012:_ExercisesOberwolfach Surgery Seminar 2012: Exercises2012-05-30T10:30:13Z<p>Martin Olbermann: /* Talks 5-7 */</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:Martin Olbermann|Martin Olbermann]]<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)]] - - [[User:Martin Olbermann|Martin Olbermann]]<br />
# [[Structures on M x I (Ex)]] - [[User:Daniel Kasprowski|Daniel Kasprowski]]<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>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Oberwolfach_Surgery_Seminar_2012:_ExercisesOberwolfach Surgery Seminar 2012: Exercises2012-05-30T10:29:18Z<p>Martin Olbermann: /* Talks 1-4 */</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:Martin Olbermann|Martin Olbermann]]<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)]] - [[User:Daniel Kasprowski|Daniel Kasprowski]]<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>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Oberwolfach_Surgery_Seminar_2012:_ExercisesOberwolfach Surgery Seminar 2012: Exercises2012-05-30T10:28:44Z<p>Martin Olbermann: /* Talks 5-7 */</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)]] - [[User:Daniel Kasprowski|Daniel Kasprowski]]<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>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Structures_on_M_x_I_(Ex)Structures on M x I (Ex)2012-05-30T09:44:28Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
Let $M$ be a closed oriented topological manifold of dimension $n \geq 5$ and let<br />
$\tilde \pi_0 \textup{Homeo}_+(M)$ denote the pseudo-isotopy group of orientation preserving self homeomorphisms of $M$: two self homeomorphisms $h_0$ and $h_1$ are pseudo-isotopic if they extend to a homeomorphism $F \colon M \times I \cong M \times I$: the group operation is composition. Let <br />
$$ \tilde \pi_0 \textup{SHomeo}_+(M) \subset \tilde \pi_0 \textup{Homeo}_+(M) $$<br />
be the subgroup of equivalences classes which are homotopic to the identity.<br />
{{beginthm|Exercise}}<br />
Using the fact that the topological surgery exact sequence is a long exact sequence of ''abelian'' groups show that $\tilde \pi_0 \textup{SHomeo}_+(M)$ is abelian.<br />
{{endthm}}<br />
</wikitex><br />
[[Category:Exercises]]<br />
[[Category:Exercises without solution]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Structures_on_M_x_I_(Ex)Structures on M x I (Ex)2012-05-30T09:34:34Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
Let $M$ be a closed oriented topological manifold of dimension $n \geq 5$ and let<br />
$\tilde \pi_0 \textup{Homeo}_+(M)$ denote the pseudo-isotopy group of orientation preserving self homeomorphisms of $M$: two self homeomorphisms $h_0$ and $h_1$ are pseudo-isotopic if then extend to a homeomorphism $F \colon M \times I \cong M \times I$: the group operation is composition. Let <br />
$$ \tilde \pi_0 \textup{SHomeo}_+(M) \subset \tilde \pi_0 \textup{Homeo}_+(M) $$<br />
be the subgroup of equivalences classes which are homotopic to the identity.<br />
{{beginthm|Exercise}}<br />
Using the fact that the topological surgery exact sequence is a long exact sequence of ''abelian'' groups show that $\tilde \pi_0 \textup{SHomeo}_+(M)$ is abelian.<br />
{{endthm}}<br />
</wikitex><br />
[[Category:Exercises]]<br />
[[Category:Exercises without solution]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Surgery_obstruction_map_I_(Ex)Talk:Surgery obstruction map I (Ex)2012-05-29T21:07:40Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
If $X$ is a manifold, then the normal map $id_X$ gives the base point of $\mathcal N (X)$.<br />
An element of $[X,G/TOP]$ is given by a bundle $\xi$ together with a fiber homotopy trivialization $\phi$.<br />
Under the isomorphism $\mathcal N (X) \cong [X,G/TOP]$, the pair $(\xi,\phi)$ corresponds to a normal map $M\to X$ covered by $\nu_M\to \nu_X\oplus \xi$. <br />
The surgery obstruction of a normal map $M\to X$ covered by $\nu_M\to \eta$ equals <br />
$$ sign(M)-sign(X)=\langle L(-\eta),[X]\rangle - \langle L(TX),[X]\rangle,$$<br />
so it depends only on the bundle over $X$.<br />
Now $(\xi\oplus\xi,\phi * \phi)$ is the sum of $(\xi,\phi)$ and $(\xi,\phi)$ in $\mathcal N (X)=[X,G/O]$ with respect to the Whitney sum.<br />
Moreover<br />
$$\theta(-(\xi,-\phi))+\theta(-(\xi,\phi)) - \theta(-(\xi\oplus\xi,\phi * \phi))<br />
= 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -\langle L(TX),[X]\rangle $$<br />
If this is non-zero, then the surgery obstruction is not a group homomorphism with respect to the Whitney sum.<br />
<br />
As an example take $X=\mathbb H P^2$:<br />
<br />
There are fiber homotopically trivial bundles on $X$ corresponding to classes in $[X,G/TOP]$<br />
which restrict to any given class in $[S^4,G/Top]$, as follows from the Puppe sequence with $\pi_7(G/TOP)=0$.<br />
From [[Fibre_homotopy_trivial_bundles_(Ex)|another exercise]] we know that on $S^4$ we have such vector bundles with first Pontryagin class $48k$ times the generator of $H^4(S^4)$.<br />
This means that on $X$ we have a vector bundle $\xi$ with $p_1(\xi)=48$ whose sphere bundle is fiber homotopically trivial, by a fiber homotopy equivalence $\phi$. <br />
We compute <br />
$$ 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -\langle L(TX),[X]\rangle<br />
= c \langle p_1(\xi)^2 ,[X] \rangle \ne 0, $$<br />
where the constant $c$ can be computed from the L-genus to be $-1/9$.<br />
<br />
So the surgery obstruction is not a group homomorphism with respect to the Whitney sum.<br />
<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Surgery_obstruction_map_I_(Ex)Talk:Surgery obstruction map I (Ex)2012-05-29T21:00:08Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We take $X=\mathbb H P^2$:<br />
The normal map $id_X$ gives the base point of $\mathcal N (X)$.<br />
An element of $[X,G/TOP]$ is given by a bundle $\xi$ together with a fiber homotopy trivialization $\phi$.<br />
Under the isomorphism $\mathcal N (X) \cong [X,G/TOP]$, the pair $(\xi,\phi)$ corresponds to a normal map $M\to X$ covered by $\nu_M\to \nu_X\oplus \xi$. <br />
The surgery obstruction of a normal map $M\to X$ covered by $\nu_M\to \eta$ equals <br />
$$ sign(M)-sign(X)=\langle L(-\eta),[X]\rangle -1,$$<br />
so it depends only on the bundle over $X$.<br />
There are fiber homotopically trivial bundles on $X$ corresponding to classes in $[X,G/TOP]$<br />
which restrict to any given class in $[S^4,G/Top]$, as follows from the Puppe sequence with $\pi_7(G/TOP)=0$.<br />
From [[Fibre_homotopy_trivial_bundles_(Ex)|another exercise]] we know that on $S^4$ we have such vector bundles with first Pontryagin class $48k$ times the generator of $H^4(S^4)$.<br />
This means that on $X$ we have a vector bundle $\xi$ with $p_1(\xi)=48$ whose sphere bundle is fiber homotopically trivial, by a fiber homotopy equivalence $\phi$. <br />
Now $(\xi\oplus\xi,\phi * \phi)$ is the sum of $(\xi,\phi)$ and $(\xi,\phi)$ in $\mathcal N (X)=[X,G/O]$ with respect to the Whitney sum.<br />
We compute <br />
$$\theta(-(\xi,-\phi))+\theta(-(\xi,\phi)) - \theta(-(\xi\oplus\xi,\phi * \phi))<br />
= 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -1<br />
= c \langle p_1(\xi)^2 ,[X] \rangle \ne 0, $$<br />
where the constant $c$ can be computed from the L-genus to be $-1/9$.<br />
<br />
So the surgery obstruction is not a group homomorphism with respect to the Whitney sum.<br />
<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Surgery_obstruction_map_I_(Ex)Talk:Surgery obstruction map I (Ex)2012-05-29T20:53:33Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We take $X=\mathbb H P^2$:<br />
The normal map $id_X$ gives the base point of $\mathcal N (X)$.<br />
An element of $[X,G/TOP]$ is given by a bundle $\xi$ together with a fiber homotopy trivialization $\phi$.<br />
Under the isomorphism $\mathcal N (X) \cong [X,G/TOP]$, the pair $(\xi,\phi)$ corresponds to a normal map $M\to X$ covered by $\nu_M\to \nu_X\oplus \xi$. <br />
The surgery obstruction of a normal map $M\to X$ covered by $\nu_M\to \eta$ equals <br />
$$ sign(M)-sign(X)=\langle L(-\eta),[X]\rangle -1,$$<br />
so it depends only on the bundle over $X$.<br />
There are fiber homotopically trivial bundles on $X$ corresponding to classes in $[X,G/TOP]$<br />
which restrict to any given class in $[S^4,G/Top]$, as follows from the Puppe sequence with $\pi_7(G/TOP)=0$.<br />
From [[Fibre_homotopy_trivial_bundles_(Ex)|another exercise]] we know that on $S^4$ we have such vector bundles with first Pontryagin class $48k$ times the generator of $H^4(S^4)$.<br />
This means that on $X$ we have a vector bundle $\xi$ with $p_1(\xi)=48$ whose sphere bundle is fiber homotopically trivial, by a fiber homotopy equivalence $\phi$. <br />
Now $(\xi\oplus\xi,\phi * \phi)$ is the sum of $(\xi,\phi)$ and $(\xi,\phi)$ in $\mathcal N (X)=[X,G/O]$ with respect to the Whitney sum.<br />
We compute <br />
$$\theta(-(\xi,-\phi))+\theta(-(\xi,\phi)) - \theta(-(\xi\oplus\xi,\phi * \phi))<br />
= 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -1<br />
= c \langle p_1(\xi)^2 ,[X] \rangle \ne 0, $$<br />
and so the surgery obstruction is not a group homomorphism with respect to the Whitney sum.<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Surgery_obstruction_map_I_(Ex)Talk:Surgery obstruction map I (Ex)2012-05-29T20:50:41Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We take $X=\mathbb H P^2$:<br />
The normal map $id_X$ gives the base point of $\mathcal N (X)$.<br />
An element of $[X,G/TOP]$ is given by a bundle $\xi$ together with a fiber homotopy trivialization $\phi$.<br />
Under the isomorphism $\mathcal N (X) \cong [X,G/TOP]$, the pair $(\xi,\phi)$ corresponds to a normal map $M\to X$ covered by $\nu_M\to \nu_X\oplus \xi$. <br />
The surgery obstruction of a normal map $M\to X$ covered by $\nu_M\to \eta$ equals <br />
$$ sign(M)-sign(X)=\langle L(-\eta),[X]\rangle -1,$$<br />
so it depends only on the bundle over $X$.<br />
There are fiber homotopically trivial bundles on $X$ corresponding to classes in $[X,G/TOP]$<br />
which restrict to any given class in $[S^4,G/Top]$, since the corresponding Atiyah-Hirzebruch spectral sequence <br />
collapses. <br />
From [[Fibre_homotopy_trivial_bundles_(Ex)|another exercise]] we know that on $S^4$ we have such vector bundles with first Pontryagin class $48k$ times the generator of $H^4(S^4)$.<br />
This means that on $X$ we have a vector bundle $\xi$ with $p_1(\xi)=48$ whose sphere bundle is fiber homotopically trivial, by a fiber homotopy equivalence $\phi$. <br />
Now $(\xi\oplus\xi,\phi * \phi)$ is the sum of $(\xi,\phi)$ and $(\xi,\phi)$ in $\mathcal N (X)=[X,G/O]$ with respect to the Whitney sum.<br />
We compute <br />
$$\theta(-(\xi,-\phi))+\theta(-(\xi,\phi)) - \theta(-(\xi\oplus\xi,\phi * \phi))<br />
= 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -1<br />
= c \langle p_1(\xi)^2 ,[X] \rangle \ne 0, $$<br />
and so the surgery obstruction is not a group homomorphism with respect to the Whitney sum.<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Surgery_obstruction_map_I_(Ex)Surgery obstruction map I (Ex)2012-05-29T20:49:52Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
Show that the surgery obstruction map<br />
$$<br />
\theta \colon \mathcal{N} (X) \rightarrow L_{n} (\Zz [\pi_1 (X)])<br />
$$<br />
is not in general a homomorphism of abelian groups, when the normal invariants are viewed as an abelian group with the group structure coming from the Whitney sum of vector bundles.<br />
<br />
Hint: in the simply connected case and $n = 4k$, find a formula for $\theta$ in terms of the $\mathcal{L}$-class. See Exercise 13.3 in {{cite|Ranicki2002}}.<br />
</wikitex><br />
== References ==<br />
{{#RefList:}}<br />
[[Category:Exercises]]<br />
[[Category:Exercises with solution]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Surgery_obstruction_map_I_(Ex)Talk:Surgery obstruction map I (Ex)2012-05-29T20:47:11Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We take $X=\mathbb H P^2$:<br />
The normal map $id_X$ gives the base point of $\mathcal N (X)$.<br />
An element of $[X,G/TOP]$ is given by a bundle $\xi$ together with a fiber homotopy trivialization $\phi$.<br />
Under the isomorphism $\mathcal N (X) \cong [X,G/TOP]$, the pair $(\xi,\phi)$ corresponds to a normal map $M\to X$ covered by $\nu_M\to \nu_X\oplus \xi$. <br />
The surgery obstruction of a normal map $M\to X$ covered by $\nu_M\to \eta$ equals <br />
$$ sign(M)-sign(X)=\langle L(-\eta),[X]\rangle -1,$$<br />
so it depends only on the bundle over $X$.<br />
There are fiber homotopically trivial bundles on $X$ corresponding to classes in $[X,G/TOP]$<br />
which restrict to any given class in $[S^4,G/Top]$, since the corresponding Atiyah-Hirzebruch spectral sequence <br />
collapses. <br />
From [[Fibre_homotopy_trivial_bundles_(Ex)|another exercise]] we know that on $S^4$ we have such vector bundles with first Pontryagin class $48k$ times the generator of $H^4(S^4)$.<br />
This means that on $X$ we have a vector bundle $\xi$ with $p_1(\xi)=48$ whose sphere bundle is fiber homotopically trivial, by a fiber homotopy equivalence $\phi$. <br />
Now $(\xi\oplus\xi,\phi * \phi)$ is the sum of $(\xi,\phi)$ and $(\xi,\phi)$ in $\mathcal N (X)=[X,G/O]$ with respect to the Whitney sum.<br />
We compute <br />
$$\sigma(-(\xi,-\phi))+\sigma(-(\xi,\phi)) - \sigma(-(\xi\oplus\xi,\phi * \phi))<br />
= 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -1<br />
= c \langle p_1(\xi)^2 ,[X] \rangle \ne 0, $$<br />
and so the surgery obstruction is not a group homomorphism with respect to the Whitney sum.<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Surgery_obstruction_map_I_(Ex)Talk:Surgery obstruction map I (Ex)2012-05-29T20:46:10Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We take $X=\mathbb H P^2$:<br />
The normal map $id_X$ gives the base point of $\mathcal N (X)\cong [X,G/TOP]$.<br />
An element of $[X,G/TOP]$ is given by a bundle $\xi$ together with a fiber homotopy trivialization $\phi$.<br />
Under the isomorphism $\mathcal N (X) \cong [X,G/O]$, the pair $(\xi,\phi)$ corresponds to a normal map $M\to X$ covered by $\nu_M\to \nu_X\oplus \xi$. <br />
The surgery obstruction of a normal map $M\to X$ covered by $\nu_M\to \eta$ equals <br />
$$ sign(M)-sign(X)=\langle L(-\eta),[X]\rangle -1,$$<br />
so it depends only on the bundle over $X$.<br />
There are fiber homotopically trivial bundles on $X$ corresponding to classes in $[X,G/TOP]$<br />
which restrict to any given class in $[S^4,G/Top]$, since the corresponding Atiyah-Hirzebruch spectral sequence <br />
collapses. <br />
From [[Fibre_homotopy_trivial_bundles_(Ex)|another exercise]] we know that on $S^4$ we have such vector bundles with first Pontryagin class $48k$ times the generator of $H^4(S^4)$.<br />
This means that on $X$ we have a vector bundle $\xi$ with $p_1(\xi)=48$ whose sphere bundle is fiber homotopically trivial, by a fiber homotopy equivalence $\phi$. <br />
Now $(\xi\oplus\xi,\phi * \phi)$ is the sum of $(\xi,\phi)$ and $(\xi,\phi)$ in $\mathcal N (X)=[X,G/O]$ with respect to the Whitney sum.<br />
We compute <br />
$$\sigma(-(\xi,-\phi))+\sigma(-(\xi,\phi)) - \sigma(-(\xi\oplus\xi,\phi * \phi))<br />
= 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -1<br />
= c \langle p_1(\xi)^2 ,[X] \rangle \ne 0, $$<br />
and so the surgery obstruction is not a group homomorphism with respect to the Whitney sum.<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Surgery_obstruction_map_I_(Ex)Talk:Surgery obstruction map I (Ex)2012-05-29T20:44:47Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We take $X=\mathbb H P^2$ and consider various bundle reductions of the normal bundle:<br />
The normal map $id_X$ gives the base point of $\mathcal N (X)\cong [X,G/TOP]$.<br />
An element of $[X,G/TOP]$ is given by a bundle $\xi$ together with a fiber homotopy trivialization $\phi$.<br />
Under the isomorphism $\mathcal N (X) \cong [X,G/O]$, the pair $(\xi,\phi)$ corresponds to a normal map $M\to X$ covered by $\nu_M\to \nu_X\oplus \xi$. <br />
The surgery obstruction of a normal map $M\to X$ covered by $\nu_M\to \eta$ equals <br />
$$ sign(M)-sign(X)=\langle L(-\eta),[X]\rangle -1,$$<br />
so it depends only on the bundle over $X$.<br />
There are fiber homotopically trivial bundles on $X$ corresponding to classes in $[X,G/TOP]$<br />
which restrict to any given class in $[S^4,G/Top]$, since the corresponding Atiyah-Hirzebruch spectral sequence <br />
collapses. <br />
From [[Fibre_homotopy_trivial_bundles_(Ex)|another exercise]] we know that on $S^4$ we have such vector bundles with first Pontryagin class $48k$ times the generator of $H^4(S^4)$.<br />
This means that on $X$ we have a vector bundle $\xi$ with $p_1(\xi)=48$ whose sphere bundle is fiber homotopically trivial, by a fiber homotopy equivalence $\phi$. <br />
Now $(\xi\oplus\xi,\phi * \phi)$ is the sum of $(\xi,\phi)$ and $(\xi,\phi)$ in $\mathcal N (X)=[X,G/O]$ with respect to the Whitney sum.<br />
We compute <br />
$$\sigma(-(\xi,-\phi))+\sigma(-(\xi,\phi)) - \sigma(-(\xi\oplus\xi,\phi * \phi))<br />
= 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -1<br />
= c \langle p_1(\xi)^2 ,[X] \rangle \ne 0, $$<br />
and so the surgery obstruction is not a group homomorphism with respect to the Whitney sum.<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Surgery_obstruction_map_I_(Ex)Talk:Surgery obstruction map I (Ex)2012-05-29T20:44:28Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We take $X=\mathbb H P^2$ and consider various bundle reductions of the normal bundle:<br />
The normal map $id_X$ gives the base point of $\mathcal N (X)\cong [X,G/TOP]$.<br />
An element of $[X,G/TOP]$ is given by a bundle $\xi$ together with a fiber homotopy trivialization $\phi$.<br />
Under the isomorphism $\mathcal N (X) \cong [X,G/O]$, the pair $(\xi,\phi)$ corresponds to a normal map $M\to X$ covered by $\nu_M\to \nu_X\oplus \xi$. <br />
The surgery obstruction of a normal map $M\to X$ covered by $\nu_M\to \eta$ equals <br />
$$ sign(M)-sign(X)=\langle L(-\eta),[X]\rangle -1,$$<br />
so it depends only on the bundle over $X$.<br />
There are fiber homotopically trivial bundles on $X$ corresponding to classes in $[X,G/TOP]$<br />
which restrict to any given class in $[S^4,G/Top]$, since the corresponding Atiyah-Hirzebruch spectral sequence <br />
collapses. <br />
From [[Fibre_homotopy_trivial_bundles_(Ex)|another exercise]] we know that on $S^4$ we have such vector bundles with first Pontryagin class $48k$ times the generator of $H^4(S^4)$.<br />
This means that on $X$ we have a vector bundle $\xi$ with $p_1(\xi)=48$ whose sphere bundle is fiber homotopically trivial, by a fiber homotopy equivalence $\phi$. <br />
Now $(\xi\oplus\xi,\phi * \phi)$ is the sum of $(\xi,\phi)$ and $(\xi,\phi)$ in $\mathcal N (X)=[X,G/O]$ with respect to the Whitney sum.<br />
We compute <br />
$$\sigma(-(\xi,-\phi))+\sigma(-(\xi,\phi)) - \sigma(-(\xi\oplus\xi,\phi * \phi))<br />
= 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -1<br />
= c \langle p_1(\xi)^2 ,[X] \rangle \ne 0$, $$<br />
and so the surgery obstruction is not a group homomorphism with respect to the Whitney sum.<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Surgery_obstruction_map_I_(Ex)Talk:Surgery obstruction map I (Ex)2012-05-29T20:28:21Z<p>Martin Olbermann: Created page with "<wikitex>; We take $X=\mathbb H P^2$ and consider various bundle reductions of the normal bundle: The normal map $id_X$ gives the base point of $\mathcal N (X)$. The surgery o..."</p>
<hr />
<div><wikitex>;<br />
We take $X=\mathbb H P^2$ and consider various bundle reductions of the normal bundle:<br />
The normal map $id_X$ gives the base point of $\mathcal N (X)$.<br />
The surgery obstruction of a normal map $M\to X$ covered by $\nu_M\to \xi$ equals <br />
$$ sign(M)-sign(X)=\langle L(-\xi),[X]\rangle -1,$$<br />
so it depends only on the bundle over $X$.<br />
There are fiber homotopically trivial bundles on $X$ corresponding to classes in $[X,G/TOP]$<br />
which restrict to any given class in $[S^4,G/Top]$, since the corresponding Atiyah-Hirzebruch spectral sequence <br />
collapses. <br />
From [[Fibre_homotopy_trivial_bundles_(Ex)|another exercise]] we know that on $S^4$ we have such vector bundles with first Pontryagin class $48k$ times the generator of $H^4(S^4)$.<br />
This means that on $X$ we have vector bundles $\xi_1,\xi_2$ whose sphere bundles are fiber homotopically trivial, by fiber homotopy equivalences $\phi_i$. Then $(\xi_1\oplus\xi_2,\phi_i * \phi_2)$ is the sum of $(\xi_1,\phi_1)$ and $(\xi_2,\phi_2)$ in<br />
$\mathcal N (X)=[X,G/O]$ with respect to the Whitney sum.<br />
Now we compute <br />
$$\sigma(\xi_1,\phi_1)+\sigma(\xi_2,\phi_2) - \sigma(\xi_1\oplus\xi_2,\phi_i * \phi_2)<br />
= \langle L(TX\oplus\xi_1), [X] \rangle + \langle L(TX\oplus\xi_2), [X] \rangle - \langle L(TX\oplus\xi_1\oplus\xi_2), [X] \rangle -\langle L(TX), [X] \rangle. $$<br />
<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Fibre_homotopy_trivial_bundles_(Ex)Talk:Fibre homotopy trivial bundles (Ex)2012-05-29T18:32:09Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We consider 5-dimensional real vector bundles over $S^4$.<br />
Isomorphism classes of these are given by their clutching function in $\pi_3(O_5)$.<br />
<br />
Given that $J:\pi_3(O_5)\to \pi_3(G_5)$ is isomorphic to the surjection $\mathbb Z\to \mathbb Z/24$,<br />
we see that the vector bundle $\xi_k$ corresponding to $24k$ times the generator<br />
has a sphere bundle $\pi:S(\xi_k)\to S^4$ which is fiber homotopically trivial, so in particular we have <br />
homotopy equivalences $f_k:S(\xi_k)\to S^4\times S^4$.<br />
<br />
From [[Tangent_bundles_of_bundles_(Ex)|another exercise]] we know that stably $TS(\xi_k)\cong \pi^*\xi_k$. <br />
From [[Obstruction_classes_and_Pontrjagin_classes_(Ex)|a third exercise]] we know that the first Pontryagin class of $\xi_k$ is $48k$ times the generator of $H^4(S^4)$. <br />
<br />
It follows that the first Pontryagin class of $S(\xi_k)$<br />
is non-trivial, since under $f_k$ the map $\pi$ just corresponds to projection to one factor.<br />
Hence $f_k$ is a homotopy equivalence which doesn't preserve the first Pontryagin class, as $S^4\times S^4$ has<br />
stably trivial tangent bundle, hence trivial $p_1$. <br />
<br />
Similarly one can argue with $(4n+1)$-dimensional vector bundles over $S^{4n}$; the $J$-homomorphism has always <br />
a non-trivial kernel, and the top Pontryagin class of the corresponding bundles are non-zero. This produces<br />
homotopy equivalences $S(\xi_k)\to S^{4n}\times S^{4n}$ which do not preserve $p_n$.<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Fibre_homotopy_trivial_bundles_(Ex)Talk:Fibre homotopy trivial bundles (Ex)2012-05-29T18:30:49Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We consider 5-dimensional real vector bundles over $S^4$.<br />
Isomorphism classes of these are given by their clutching function in $\pi_3(O_5)$.<br />
<br />
Given that $J:\pi_3(O_5)\to \pi_3(G_5)$ is isomorphic to the surjection $\mathbb Z\to \mathbb Z/24$,<br />
we see that the vector bundle $\xi_k$ corresponding to $24k$ times the generator<br />
has a sphere bundle $\pi:S(\xi_k)\to S^4$ which is fiber homotopically trivial, so in particular we have <br />
homotopy equivalences $f_k:S(\xi_k)\to S^4\times S^4$.<br />
<br />
From [[Tangent_bundles_of_bundles_(Ex)|another exercise]] we know that stably $TS(\xi_k)\cong \pi^*\xi_k$. <br />
From [[Obstruction_classes_and_Pontrjagin_classes_(Ex)|a third exercise]] we know that the first Pontryagin class of $\xi_k$ is $48k$. <br />
<br />
It follows that the first Pontryagin class of $S(\xi_k)$<br />
is non-trivial, since under $f_k$ the map $\pi$ just corresponds to projection to one factor.<br />
Hence $f_k$ is a homotopy equivalence which doesn't preserve the first Pontryagin class, as $S^4\times S^4$ has<br />
stably trivial tangent bundle, hence trivial $p_1$. <br />
<br />
Similarly one can argue with $(4n+1)$-dimensional vector bundles over $S^{4n}$; the $J$-homomorphism has always <br />
a non-trivial kernel, and the top Pontryagin class of the corresponding bundles are non-zero. This produces<br />
homotopy equivalences $S(\xi_k)\to S^{4n}\times S^{4n}$ which do not preserve $p_n$.<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Fibre_homotopy_trivial_bundles_(Ex)Fibre homotopy trivial bundles (Ex)2012-05-29T18:29:32Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
{{beginthm|Exercise}}<br />
# Observe that $\pi_i(G_{k+1})$ classifies $k$-spherical fibrations over $S^{i+1}$. Using the isomorphisms $\pi_3(G_5) \cong \pi_3^S \cong \Zz/24$, $\pi_3(O_5) \cong \Zz$ and the fact that the J-homomophism in dimension $3$ is isomorphic to the surjective homomorphism $\Zz \to \Zz/24$, find a homotopy equivalence of manifolds $f \colon M_0 \simeq M_1$ such that $f^*p(M_0) \neq p(M_1)$. Here $p(M_i) \in H^{4*}(M_i; \Zz)$ denotes the total Pontrjagin class.<br />
# The above exercise showed that the first Pontrjain class, $p_1$, is not a homotopy invariant. Apply the same idea to show that $p_k$ is not a homotopy invariant for any $k \geq 1$.<br />
{{endthm}}<br />
{{beginrem|Remark}} A reference to Novikov is needed here.<br />
{{endrem}}<br />
</wikitex><br />
== References ==<br />
{{#RefList:}}<br />
[[Category:Exercises]]<br />
[[Category:Exercises with solution]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Fibre_homotopy_trivial_bundles_(Ex)Talk:Fibre homotopy trivial bundles (Ex)2012-05-29T18:21:41Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We consider 5-dimensional real vector bundles over $S^4$.<br />
Isomorphism classes of these are given by their clutching function in $\pi_3(O_5)$.<br />
<br />
Given that $J:\pi_3(O_5)\to \pi_3(G_5)$ is isomorphic to the surjection $\mathbb Z\to \mathbb Z/24$,<br />
we see that the vector bundle $\xi_k$ corresponding to $24k$ times the generator<br />
has a sphere bundle $\pi:S(\xi_k)\to S^4$ which is fiber homotopically trivial, so in particular we have <br />
homotopy equivalences $f_k:S(\xi_k)\to S^4\times S^4$.<br />
<br />
From another [[Tangent_bundles_of_bundles_(Ex)|exercise]] we know that stably $TS(\xi_k)\cong \pi^*\xi_k$. <br />
From a third [[Obstruction_classes_and_Pontrjagin_classes_(Ex)|exercise]] we know that the first Pontryagin class of $\xi_k$ is $48k$. <br />
<br />
It follows that the first Pontryagin class of $S(\xi_k)$<br />
is non-trivial, since under $f_k$ the map $\pi$ just corresponds to projection to one factor.<br />
Hence $f_k$ is a homotopy equivalence which doesn't preserve the first Pontryagin class, as $S^4\times S^4$ has<br />
stably trivial tangent bundle, hence trivial $p_1$. <br />
<br />
Similarly one can argue with $(4n+1)$-dimensional vector bundles over $S^{4n}$; the $J$-homomorphism has always <br />
a non-trivial kernel, and the top Pontryagin class of the corresponding bundles are non-zero. This produces<br />
homotopy equivalences $S(\xi_k)\to S^{4n}\times S^{4n}$ which do not preserve $p_n$.<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Fibre_homotopy_trivial_bundles_(Ex)Talk:Fibre homotopy trivial bundles (Ex)2012-05-29T18:17:32Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We consider 5-dimensional real vector bundles over $S^4$.<br />
Isomorphism classes of these are given by their clutching function in $\pi_3(O_5)$.<br />
<br />
Given that $J:\pi_3(O_5)\to \pi_3(G_5)$ is isomorphic to the surjection $\mathbb Z\to \mathbb Z/24$,<br />
we see that the vector bundle $\xi_k$ corresponding to $24k$ times the generator<br />
has a sphere bundle $\pi:S(\xi_k)\to S^4$ which is fiber homotopically trivial, so in particular we have <br />
homotopy equivalences $f_k:S(\xi_k)\to S^4\times S^4$.<br />
<br />
From another exercise we know that stably $TS(\xi_k)\cong \pi^*\xi_k$. <br />
From a third exercise we know that the first Pontryagin class of $\xi_k$ is $48k$. <br />
<br />
It follows that the first Pontryagin class of $S(\xi_k)$<br />
is non-trivial, since under $f_k$ the map $\pi$ just corresponds to projection to one factor.<br />
Hence $f_k$ is a homotopy equivalence which doesn't preserve the first Pontryagin class, as $S^4\times S^4$ has<br />
stably trivial tangent bundle, hence trivial $p_1$. <br />
<br />
Similarly one can argue with $(4n+1)$-dimensional vector bundles over $S^{4n}$; the $J$-homomorphism has always <br />
a non-trivial kernel, and the top Pontryagin class of the corresponding bundles are non-zero. This produces<br />
homotopy equivalences $S(\xi_k)\to S^{4n}\times S^{4n}$ which do not preserve $p_n$.<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Fibre_homotopy_trivial_bundles_(Ex)Talk:Fibre homotopy trivial bundles (Ex)2012-05-29T18:16:08Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex>;<br />
We consider 5-dimensional real vector bundles over $S^4$.<br />
Isomorphism classes of these are given by their clutching function in $\pi_3(O_5)$<br />
Given that $J:\pi_3(O_5)\to \pi_3(G_5)$ is isomorphic to the surjection $\mathbb Z\to \mathbb Z/24$,<br />
we see that the vector bundle $\xi_k$ corresponding to $24k$ times the generator<br />
has a sphere bundle $\pi:S(\xi_k)\to S^4$ which is fiber homotopically trivial, so in particular we have <br />
homotopy equivalences $f_k:S(\xi_k)\to S^4\times S^4$.<br />
From another exercise we know that stably $TS(\xi_k)\cong \pi^*\xi_k$. From a third exercise we know that <br />
the first Pontryagin class of $xi_k$ is $48k$. It follows that the first Pontryagin class of $S(\xi_k)$<br />
is non-trivial, since under $f_k$ the map $pi$ just corresponds to projection to one factor.<br />
Hence $f_k$ is a homotopy equivalence which doesn't preserve the first Pontryagin class, as $S^4\times S^4$ has<br />
stably trivial tangent bundle, hence trivial $p_1$. <br />
<br />
Similarly one can argue with $(4n+1)$-dimensional vector bundles over $S^{4n}$; the J-homomorphism has always <br />
a non-trivial kernel, and the top Pontryagin class of the corresponding bundles are non-zero. This produces<br />
homotopy equivalences $S(\xi_k)\to S^{4n}\times S^{4n}$ which do not preserve $p_n$.<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Fibre_homotopy_trivial_bundles_(Ex)Talk:Fibre homotopy trivial bundles (Ex)2012-05-29T18:01:13Z<p>Martin Olbermann: </p>
<hr />
<div><wikitex><br />
We consider 5-dimensional real vector bundles over $S^4$.<br />
Isomorphism classes of these are given by their clutching function in $\pi_3(O_5)$<br />
Given that $\pi_3(O_5)\to \pi_3(G_5)$ is isomorphic to the surjection $\mathbb Z\to \mathbb Z/24$,<br />
we see that the vector bundle $\xi_k$ corresponding to $24k$ times the generator<br />
has a sphere bundle $S(xik)$ which is fiber homotopically trivial, so in particular we have <br />
homotopy equivalences <br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Talk:Fibre_homotopy_trivial_bundles_(Ex)Talk:Fibre homotopy trivial bundles (Ex)2012-05-29T17:59:19Z<p>Martin Olbermann: Created page with "<wikitex> We consider 5-dimensional real vector bundles over $S^4$. Isomorphism classes of these are given by their clutching function in $\pi_3(O_5)$ Given that $\pi_3(O_5)\t..."</p>
<hr />
<div><wikitex><br />
We consider 5-dimensional real vector bundles over $S^4$.<br />
Isomorphism classes of these are given by their clutching function in $\pi_3(O_5)$<br />
Given that $\pi_3(O_5)\to \pi_3(G_5)$ is isomorphic to the surjection $\Z\to \Z/24$,<br />
we see that the vector bundle $\xi_k$ corresponding to $24k$ times the generator<br />
has a sphere bundle $S(xik$<br />
</wikitex></div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/String_bordismString bordism2011-04-01T15:22:04Z<p>Martin Olbermann: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
$String$-bordism or $O\!\left< 8 \right>$-bordism is a special case of a [[B-Bordism|B-bordism]]. It comes from the tower of fibrations below.<br />
$$<br />
\xymatrix{BO\!\left< 8 \right>\ar[d] & \\ BSpin \ar[d]\ar[r]^{p_1/2}&K({\mathbb Z},4) \\ BSO\ar[d]\ar[r]^{w_2}& K({\mathbb Z}/2,2) \\ BO\ar[r]^{w_1}&<br />
K({\mathbb Z}/2,1) }<br />
$$<br />
In each step the lowest homotopy group is killed by the map into the Eilenberg-MacLane spaces. In particular, $BO\left< 8 \right>$ is the homotopy fibre of the map from $BSpin$ given by half of the first Pontryagin class. The name $String$-group is due to Haynes Miller and will be explained below.<br />
<br />
</wikitex><br />
<br />
== The String group ==<br />
<wikitex>;<br />
There are various models for the String group. However, since its first three homotopy groups vanish it can not be realized as a finite dimensional Lie group. There are homotopy theoretic constructions which involve path spaces. A more geometric way to think about $String(n)$ which involves only finite dimensional manifolds is the following: the String group fibers over the Spin group with fibre $K({\mathbb Z},2)$. One may think of $ K({\mathbb Z},2)$ as the realization of $S^1$ viewed as a smooth category with only one object. This way, the $A_\infty$ space $String(n)$ appears as the realization of a smooth 2-group extension of $Spin(n)$ by the finite dimensional Lie groupoid $S^1$ (see \cite{Schommer-Pries2009}). A more explicit model for this extension can be found in \cite{Meinrenken2003}. <br />
</wikitex><br />
<br />
== The bordism groups ==<br />
<wikitex>;<br />
The additive structure of the bordism groups is not fully determined yet. It is known that only 2 and 3 torsion appears (see \cite{Giambalvo1971}) and the 3 torsion is annihalated by multiplication with 3 (see \cite{Hovey1997}). Moreover, the bordism groups $\Omega_{k}^{String}$ are finite for $k=1,2,3$ mod 4.<br />
<br />
Clearly, since $BO\!\left< 8 \right>$ is 7-connected the first 6 bordism groups coincide with the [[Framed bordism|framed bordism groups]]. The first 16 bordism groups have been computed by Giambalvo \cite{Giambalvo1971|p. 538}:<br />
* $\Omega_7^{String} = 0$.<br />
* $\Omega_8^{String} \cong \Zz \oplus \Zz/2$, generated by the [[Exotic spheres|exotic 8-sphere]] for the 2-torsion and a certain [[Bott manifold]]: see \cite{Laures2004}.<br />
* $\Omega_9^{String} \cong \Theta_9/bP_{10} \cong (\Zz/2)^2$, generated by [[Exotic spheres|exotic 9-spheres]].<br />
* $\Omega_{10}^{String} \cong \Zz/6$, generated by an [[Exotic spheres|exotic 10-sphere]].<br />
* $\Omega_{11}^{String} = 0$.<br />
* $\Omega_{12}^{String} \cong \Zz$, generated by a 5-connected manifold with signature $8 \times 992$.<br />
* $\Omega_{13}^{String} = 0$.<br />
* $\Omega_{14}^{String} \cong \Zz/2$, generated by the [[Exotic spheres|exotic 14-sphere]].<br />
* $\Omega_{15}^{String} \cong \Zz/2$, genreated by the [[Exotic spheres|exotic 15-sphere]].<br />
* $\Omega_{16}^{String} \cong \Zz^2$.<br />
At the prime 3 the first 32 bordism groups can be found in \cite{Hovey&Ravenel1995}. Further calculations have been done in \cite{Mahowald&Gorbounov1995}. <br />
</wikitex><br />
<br />
== Homology calculations ==<br />
===Singular homology===<br />
<wikitex>;<br />
The cohomology ring $H^*(BString,{\mathbb Z}/p)$ has been computed for $p=2$ by Stong in \cite{Stong1963}: <br />
$$ H^*(BString)\cong {\mathbb Z}/2[\theta_i|\sigma_2(i+1)>4]\otimes H^*(K({\mathbb Z},4))/Sq^2(\iota).$$<br />
Here, $\sigma_2$ is the number of digits in the duadic decomposition and the $\theta_i$ come from the cohomology of $BO$ and coincide with the Stiefel-Whitney up to decomposables. For odd $p$ the corresponding result has been obtained by Giambalvo \cite{Giambalvo1969}.<br />
<br />
</wikitex><br />
<br />
===K(1)-local computations===<br />
<wikitex>;<br />
$K(1)$ locally $MString$ coincides with $MSpin$ and decomposes into a wedge of copies of $KO$. However, it is not an algebra over $KO$. Its multiplicative structure for $p=2$ can be read off the formula<br />
$$ L_{K(1)}MString \cong T_\zeta \wedge \bigwedge TS^0.$$<br />
Here, $\zeta\in \pi_{-1}L_{K(1)}S^0$ is a generator, $T_\zeta$ is the $E_\infty $ cone over $\zeta$ and $TS^0$ is the free $E_\infty$ spectrum generated by the sphere. In particular, its $\theta$-algebra structure is free (see \cite{Laures2003a}).<br />
<br />
</wikitex><br />
<br />
===K(n)-local computations===<br />
<br />
<wikitex>;<br />
For $K=K(n)$ at $p=2$ one has an exact sequence of Hopf algebras (see \cite{Kitchloo&Laures&Wilson2004a})<br />
$$ \xymatrix{K_* \ar[r] & K_*K({\mathbb Z}/2,2)\ar[r]& K_*K({\mathbb Z},3)\ar[r]& K_*BString \ar[r]&K_*BSpin \ar[r]& K_* K({\mathbb Z},4)\ar[r]^2& K({\mathbb Z},4)\ar[r]&K_*}$$<br />
which is induced by the obvious geometric maps. For $n=2$ it algebraically reduces to the split exact sequence of Hopf algebras (see \cite{Kitchloo&Laures&Wilson2004b})<br />
$$\xymatrix{K(2)_* \ar[r] & K(2)_*K({\mathbb Z},3)\ar[r]& K(2)_*BString \ar[r]&K(2)_*BSpin \ar[r]& K(2)_*}.$$<br />
</wikitex><br />
<br />
===Computations with respect to general complex oriented theories===<br />
<wikitex>;<br />
Ando, Hopkins and Strickland investigated the homology ring $E_*BString$ for even periodic multiplicative cohomology theories $E$. Even periodic theories are complex orientable which means that $E^0({\mathbb C}P^\infty)$ carries a formal group. The description of $E_*BString$ is in terms of formal group data. <br />
<br />
In \cite{Ando&Hopkins&Strickland2001a} first the analogous complex problem is studied. The group $O\left< 8 \right>=String$ has a complex relative $U\left< 6 \right>$ which is defined in the same way by killing the third homotopy group of $SU$. Consider the map<br />
$$ (1-L_1)(1-L_2)(1-L_3):<br />
\xymatrix{ {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty\ar[r]& BU}$$<br />
where the $L_i$ are the canonical line bundles over the individual factors. Since the first two Chern classes of the virtual bundle vanish the map lifts to $BU\left<6 \right>$. If we choose a complex orienatation the lift gives a class $f$ in the cohomology <br />
ring<br />
$$(E\wedge BU\left<6 \right>_+)^0( {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty )\cong <br />
E_0BU\left<6 \right> [\![x_1,x_2,x_3]\!]$$ <br />
with $c_i(L_i)=x_i$. The power series $f$ satisfies the following identities:<br />
$$ \begin{aligned}<br />
f(0,0,0)&=1\\<br />
f(x_1,x_2,x_3)&=f(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)});\; \sigma \in \Sigma_3\\<br />
f(x_1,x_2,x_3)f(x_0,x_1+_Fx_2,x_3)&=f(x_0+_Fx_1,x_2,x_3)f(x_0,x_1,x_2).<br />
\end{aligned}$$<br />
Hence, the power series is an example a symmetric 3-cocycle or a cubical structure on the formal group law. In fact the main result of \cite{Ando&Hopkins&Strickland2001a} is that it is the universal example of such a structure. Explicitly, this means that the commutative ring $E_0 BU\left <6 \right>$ is freely generated by the coefficients of $f$ subject to the relations given by the 3 equations above.<br />
<br />
The real version of this result has not been published yet by the three authors. Using the diagram<br />
$$ \xymatrix{K{\mathbb Z},3)\ar[r]\ar[d]^=&BU\left< 6 \right>\ar[r]\ar[d]& BSU\ar[d]\\<br />
K({\mathbb Z},3)\ar[r]& BString \ar[r]&BSpin}$$<br />
and the results for $K(2)_*BString$ described above they conjecture that $E_0BString$ is the same quotient subject to the additional relation<br />
$$ f(x_1,x_2,x_3)=1 \mbox{ for } x_1+_Fx_2+_Fx_3=0.$$<br />
</wikitex><br />
<br />
== The structure of the spectrum==<br />
<wikitex>;<br />
Localized at a prime $p>3$, string bordism splits additively into a sum of suspensions of $BP$, although the ring structure is different (see \cite{Hovey2008}). For $p=3$ there is a spectrum $Y$ with 3 cells in even dimensions such that $MString\wedge Y$ splits into a sum of suspensions of $BP$. For $p=2$ it is hoped that the spectrum $tmf$ splits off which is explained below. <br />
<br />
</wikitex><br />
<br />
== The Witten genus ==<br />
<wikitex>;<br />
At the end of the 80s Ed Witten were studying the $S^1$-equivariant index of the Dirac operator on a loop space of a $4k$-dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space $LM$ this would mean that $M$ is $String$. Witten carried the Atiyah-Segal formula for the index over to this infinite dimensional setting and obtained an integral modular form of weight $k$. Nowadays this is called the Witten genus (see \cite{Segal1988}.)<br />
The Witten genus can be refined to a map of structured ring spectra<br />
$$W: MString \longrightarrow TMF$$<br />
from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms (\cite{Hopkins2002}). This map is also called the $\sigma$-orientation and is 15-connected (see \cite{Hill2008}). The spectrum $TMF$ was developed by Goerss, Hopkins and Miller. It is supposed to play the same role for $String$-bordism as $KO$-theory does for $Spin$-bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map $W$ gives characteristic numbers which together with $KO$ and Stiefel-Whitney numbers are conjectured to determine the $String$ bordism class. Moreover, $tmf$ is supposed to be a direct summand of $MString$ as the orientation map $W$ is shown to be surjective in homotopy (see \cite{Hopkins&Mahowald2002}.)<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]<br />
[[Category:Bordism]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/String_bordismString bordism2011-04-01T15:19:55Z<p>Martin Olbermann: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
$String$-bordism or $O\!\left< 8 \right>$-bordism is a special case of a [[B-Bordism|B-bordism]]. It comes from the tower of fibrations below.<br />
$$<br />
\xymatrix{BO\!\left< 8 \right>\ar[d] & \\ BSpin \ar[d]\ar[r]^{p_1/2}&K({\mathbb Z},4) \\ BSO\ar[d]\ar[r]^{w_2}& K({\mathbb Z}/2,2) \\ BO\ar[r]^{w_1}&<br />
K({\mathbb Z}/2,1) }<br />
$$<br />
In each step the lowest homotopy group is killed by the map into the Eilenberg-MacLane spaces. In particular, $BO\left< 8 \right>$ is the homotopy fibre of the map from $BSpin$ given by half of the first Pontryagin class. The name $String$-group is due to Haynes Miller and will be explained below.<br />
<br />
</wikitex><br />
<br />
== The String group ==<br />
<wikitex>;<br />
There are various models for the String group. However, since its first three homotopy groups vanish it can not be realized as a finite dimensional Lie group. There are homotopy theoretic constructions which involve path spaces. A more geometric way to think about $String(n)$ which involves only finite dimensional manifolds is the following: the String group fibers over the Spin group with fibre $K({\mathbb Z},2)$. One may think of $ K({\mathbb Z},2)$ as the realization of $S^1$ viewed as a smooth category with only one object. This way, the $A_\infty$ space $String(n)$ appears as the realization of a smooth 2-group extension of $Spin(n)$ by the finite dimensional Lie groupoid $S^1$ (see \cite{Schommer-Pries2009}). A more explicit model for this extension can be found in \cite{Meinrenken2003}. <br />
</wikitex><br />
<br />
== The bordism groups ==<br />
<wikitex>;<br />
The additive structure of the bordism groups is not fully determined yet. It is known that only 2 and 3 torsion appears (see \cite{Giambalvo1971}) and the 3 torsion is annihalated by multiplication with 3 (see \cite{Hovey1997}). Moreover, the bordism groups $\Omega_{k}^{String}$ are finite for $k=1,2,3$ mod 4.<br />
<br />
Clearly, since $BO\!\left< 8 \right>$ is 7-connected the first 6 bordism groups coincide with the [[Framed bordism|framed bordism groups]]. The first 16 bordism groups have been computed by Giambalvo \cite{Giambalvo1971|p. 538}:<br />
* $\Omega_7^{String} = 0$.<br />
* $\Omega_8^{String} \cong \Zz \oplus \Zz/2$, generated by the [[Exotic spheres|exotic 8-sphere]] for the 2-torsion and a certain [[Bott manifold]]: see \cite{Laures2004}.<br />
* $\Omega_9^{String} \cong \Theta_9/bP_{10} \cong (\Zz/2)^2$, generated by [[Exotic spheres|exotic 9-spheres]].<br />
* $\Omega_{10}^{String} \cong \Zz/6$, generated by an [[Exotic spheres|exotic 10-sphere]].<br />
* $\Omega_{11}^{String} = 0$.<br />
* $\Omega_{12}^{String} \cong \Zz$, generated by a 5-connected manifold with signature $8 \times 992$.<br />
* $\Omega_{13}^{String} = 0$.<br />
* $\Omega_{14}^{String} \cong \Zz/2$, generated by the [[Exotic spheres|exotic 14-sphere]].<br />
* $\Omega_{15}^{String} \cong \Zz/2$, genreated by the [[Exotic spheres|exotic 15-sphere]].<br />
* $\Omega_{16}^{String} \cong \Zz^2$.<br />
At the prime 3 the first 32 bordism groups can be found in \cite{Hovey&Ravenel1995}. Further calculations have been done in \cite{Mahowald&Gorbounov1995}. <br />
</wikitex><br />
<br />
== Homology calculations ==<br />
===Singular homology===<br />
<wikitex>;<br />
The cohomology ring $H^*(BString,{\mathbb Z}/p)$ has been computed for $p=2$ by Stong in \cite{Stong1963a}: <br />
$$ H^*(BString)\cong {\mathbb Z}/2[\theta_i|\sigma_2(i+1)>4]\otimes H^*(K({\mathbb Z},4))/Sq^2(\iota).$$<br />
Here, $\sigma_2$ is the number of digits in the duadic decomposition and the $\theta_i$ come from the cohomology of $BO$ and coincide with the Stiefel-Whitney up to decomposables. For odd $p$ the corresponding result has been obtained by Giambalvo \cite{Giambalvo1969}.<br />
<br />
</wikitex><br />
<br />
===K(1)-local computations===<br />
<wikitex>;<br />
$K(1)$ locally $MString$ coincides with $MSpin$ and decomposes into a wedge of copies of $KO$. However, it is not an algebra over $KO$. Its multiplicative structure for $p=2$ can be read off the formula<br />
$$ L_{K(1)}MString \cong T_\zeta \wedge \bigwedge TS^0.$$<br />
Here, $\zeta\in \pi_{-1}L_{K(1)}S^0$ is a generator, $T_\zeta$ is the $E_\infty $ cone over $\zeta$ and $TS^0$ is the free $E_\infty$ spectrum generated by the sphere. In particular, its $\theta$-algebra structure is free (see \cite{Laures2003a}).<br />
<br />
</wikitex><br />
<br />
===K(n)-local computations===<br />
<br />
<wikitex>;<br />
For $K=K(n)$ at $p=2$ one has an exact sequence of Hopf algebras (see \cite{Kitchloo&Laures&Wilson2004a})<br />
$$ \xymatrix{K_* \ar[r] & K_*K({\mathbb Z}/2,2)\ar[r]& K_*K({\mathbb Z},3)\ar[r]& K_*BString \ar[r]&K_*BSpin \ar[r]& K_* K({\mathbb Z},4)\ar[r]^2& K({\mathbb Z},4)\ar[r]&K_*}$$<br />
which is induced by the obvious geometric maps. For $n=2$ it algebraically reduces to the split exact sequence of Hopf algebras (see \cite{Kitchloo&Laures&Wilson2004b})<br />
$$\xymatrix{K(2)_* \ar[r] & K(2)_*K({\mathbb Z},3)\ar[r]& K(2)_*BString \ar[r]&K(2)_*BSpin \ar[r]& K(2)_*}.$$<br />
</wikitex><br />
<br />
===Computations with respect to general complex oriented theories===<br />
<wikitex>;<br />
Ando, Hopkins and Strickland investigated the homology ring $E_*BString$ for even periodic multiplicative cohomology theories $E$. Even periodic theories are complex orientable which means that $E^0({\mathbb C}P^\infty)$ carries a formal group. The description of $E_*BString$ is in terms of formal group data. <br />
<br />
In \cite{Ando&Hopkins&Strickland2001a} first the analogous complex problem is studied. The group $O\left< 8 \right>=String$ has a complex relative $U\left< 6 \right>$ which is defined in the same way by killing the third homotopy group of $SU$. Consider the map<br />
$$ (1-L_1)(1-L_2)(1-L_3):<br />
\xymatrix{ {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty\ar[r]& BU}$$<br />
where the $L_i$ are the canonical line bundles over the individual factors. Since the first two Chern classes of the virtual bundle vanish the map lifts to $BU\left<6 \right>$. If we choose a complex orienatation the lift gives a class $f$ in the cohomology <br />
ring<br />
$$(E\wedge BU\left<6 \right>_+)^0( {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty )\cong <br />
E_0BU\left<6 \right> [\![x_1,x_2,x_3]\!]$$ <br />
with $c_i(L_i)=x_i$. The power series $f$ satisfies the following identities:<br />
$$ \begin{aligned}<br />
f(0,0,0)&=1\\<br />
f(x_1,x_2,x_3)&=f(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)});\; \sigma \in \Sigma_3\\<br />
f(x_1,x_2,x_3)f(x_0,x_1+_Fx_2,x_3)&=f(x_0+_Fx_1,x_2,x_3)f(x_0,x_1,x_2).<br />
\end{aligned}$$<br />
Hence, the power series is an example a symmetric 3-cocycle or a cubical structure on the formal group law. In fact the main result of \cite{Ando&Hopkins&Strickland2001a} is that it is the universal example of such a structure. Explicitly, this means that the commutative ring $E_0 BU\left <6 \right>$ is freely generated by the coefficients of $f$ subject to the relations given by the 3 equations above.<br />
<br />
The real version of this result has not been published yet by the three authors. Using the diagram<br />
$$ \xymatrix{K{\mathbb Z},3)\ar[r]\ar[d]^=&BU\left< 6 \right>\ar[r]\ar[d]& BSU\ar[d]\\<br />
K({\mathbb Z},3)\ar[r]& BString \ar[r]&BSpin}$$<br />
and the results for $K(2)_*BString$ described above they conjecture that $E_0BString$ is the same quotient subject to the additional relation<br />
$$ f(x_1,x_2,x_3)=1 \mbox{ for } x_1+_Fx_2+_Fx_3=0.$$<br />
</wikitex><br />
<br />
== The structure of the spectrum==<br />
<wikitex>;<br />
Localized at a prime $p>3$, string bordism splits additively into a sum of suspensions of $BP$, although the ring structure is different (see \cite{Hovey2008}). For $p=3$ there is a spectrum $Y$ with 3 cells in even dimensions such that $MString\wedge Y$ splits into a sum of suspensions of $BP$. For $p=2$ it is hoped that the spectrum $tmf$ splits off which is explained below. <br />
<br />
</wikitex><br />
<br />
== The Witten genus ==<br />
<wikitex>;<br />
At the end of the 80s Ed Witten were studying the $S^1$-equivariant index of the Dirac operator on a loop space of a $4k$-dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space $LM$ this would mean that $M$ is $String$. Witten carried the Atiyah-Segal formula for the index over to this infinite dimensional setting and obtained an integral modular form of weight $k$. Nowadays this is called the Witten genus (see \cite{Segal1988}.)<br />
The Witten genus can be refined to a map of structured ring spectra<br />
$$W: MString \longrightarrow TMF$$<br />
from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms (\cite{Hopkins2002}). This map is also called the $\sigma$-orientation and is 15-connected (see \cite{Hill2008}). The spectrum $TMF$ was developed by Goerss, Hopkins and Miller. It is supposed to play the same role for $String$-bordism as $KO$-theory does for $Spin$-bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map $W$ gives characteristic numbers which together with $KO$ and Stiefel-Whitney numbers are conjectured to determine the $String$ bordism class. Moreover, $tmf$ is supposed to be a direct summand of $MString$ as the orientation map $W$ is shown to be surjective in homotopy (see \cite{Hopkins&Mahowald2002}.)<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]<br />
[[Category:Bordism]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Dold_manifoldDold manifold2011-04-01T14:56:23Z<p>Martin Olbermann: </p>
<hr />
<div><!-- COMMENT: <br />
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<br />
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<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
--><br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
A Dold manifold is a manifold of the form <br />
$$<br />
P(m,n):= (S^m \times \mathbb {CP}^n)/\tau,<br />
$$ where $m>0$, and the involution $\tau$ sends $(x,[y]) $ to $(-x, [\bar y])$ where $\bar y = (\bar y_0,...,\bar y_n)$ for $y =(y_0,...y_n)$.<br />
<br />
<br />
Dold used these manifolds in {{cite|Dold1956}} as generators for the [[Unoriented bordism|unoriented bordism ring]].<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
The fibre bundle $p:P(m,n) \to \mathbb {RP}^m$ has a section $s([x]) := [(x,[1,...,1])]$ and we consider the cohomology classes (always with $\mathbb Z/2$-coefficients)<br />
$$<br />
c:= p^*(x) \in H^1(P(m,n)),<br />
$$<br />
where $x$ is a generator of $H^1(\mathbb {RP}^m)$, and<br />
$$<br />
d \in H^2(P(m,n)),<br />
$$<br />
which is characterized by the property that the restriction to a fibre is non-trivial and $s^*(d)=0$.<br />
<br />
{{beginthm|Theorem {{cite|Dold1956}}}} The classes $c \in H^1(P(m,n))$ and $d\in H^2(P(m,n))$ generate $H^*(P(m,n);\mathbb Z/2)$ with only the relations<br />
$$ c^{m+1} =0<br />
$$<br />
and <br />
$$<br />
d^{n+1} =0.<br />
$$<br />
The Steenrod squares act by<br />
$$<br />
Sq^0 =id, \,\, Sq^1(c) = c^2,\,\, Sq^1(d) = cd,\,\, Sq^2(d) =d^2,<br />
$$<br />
and all other Squares $Sq^i$ act trivially on $c$ and $d$. On the decomposable classes the action is given by the Cartan formula.<br />
<br />
The total Stiefel-Whitney class of the tangent bundle is<br />
$$<br />
w(P(m,n)) = (1+c)^{m+1}(1+d)^{n+1}.<br />
$$<br />
{{endthm}}<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]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Dold_manifoldDold manifold2011-04-01T14:52:08Z<p>Martin Olbermann: Created page with "<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - Fo..."</p>
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<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 />
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<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
--><br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
A Dold manifold is a manifold of the form <br />
$$<br />
P(m,n):= (S^m \times \mathbb {CP}^n)/\tau,<br />
$$ where $m>0$, and the involution $\tau$ sends $(x,[y]) $ to $(-x, [\bar y])$ where $\bar y = (\bar y_0,...,\bar y_n)$ for $y =(y_0,...y_n)$.<br />
<br />
<br />
Dold used these manifolds in {{cite|Dold1956}} as generators for the [[Unoriented bordism|unoriented bordism ring]].<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
{{beginthm|Theorem {{cite|Dold1956}}}} The classes $c \in H^1(P(m,n))$ and $d\in H^2(P(m,n))$ generate $H^*(P(m,n);\mathbb Z/2)$ with only the relations<br />
$$ c^{m+1} =0<br />
$$<br />
and <br />
$$<br />
d^{n+1} =0.<br />
$$<br />
The Steenrod squares act by<br />
$$<br />
Sq^0 =id, \,\, Sq^1(c) = c^2,\,\, Sq^1(d) = cd,\,\, Sq^2(d) =d^2,<br />
$$<br />
and all other Squares $Sq^i$ act trivially on $c$ and $d$. On the decomposable classes the action is given by the Cartan formula.<br />
<br />
The total Stiefel-Whitney class of the tangent bundle is<br />
$$<br />
w(P(m,n)) = (1+c)^{m+1}(1+d)^{n+1}.<br />
$$<br />
{{endthm}}<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]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Manifold_Atlas:Community_PortalManifold Atlas:Community Portal2011-04-01T09:53:44Z<p>Martin Olbermann: </p>
<hr />
<div>{{DISPLAYTITLE:{{NAMESPACE}}: {{PAGENAME}}}}<br />
Welcome to the Manifold Atlas Community Portal. Below you will find a list of planned pages on the Atlas - please feel free to get involved and start working on these pages or another page you would like to create.<br />
<br />
Here are some pages you may like to visit:<br />
<br />
* [[Manifold Atlas:Projects|Projects]] you can join.<br />
<br />
* [[Manifold Atlas:Diary|Atlas diary]] for recent events as well as the history of the Atlas.<br />
<br />
* [[Manifold Atlas:The Atlas Observer|Atlas Observer]] lets you sign up for email up-dates about changes at the Atlas.<br />
<br />
* [[Manifold Atlas:Workshops|Workshops]] gives information about Atlas writing workshops.<br />
<br />
* [[Manifold Atlas:Links|Links]] for links to resources about manifolds on the Web.<br />
<br />
== Proposed pages ==<br />
In the first phase of the creation of the Manifold Atlas, we are looking for a broad range of pages. <br />
<br />
* Below are lists of pages we would like to see on the Manifold Atlas. <br />
<br />
* Please feel free to start a page from one of the lists and to add to the discussion page.<br />
<!-- == Pages under construction ==<br />
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A [{{fullurl:Special:WhatLinksHere}}?target=Template:Stub&namespace=0 list] of pages <br />
which contain the [[Template:Stub|“Stub”]] template. --><br />
<br />
* For comparison here is the [[Wikipedia:List_of_manifolds|Wikipedia list of manifolds]] and a list of [[Manifold Atlas:Wikipedia pages|Wikipedia pages related to the theory of manifolds]].<br />
<br />
=== Manifolds ===<br />
* [[Abelian varieties]]<br />
* [[Arithmetic 3-manifolds]]<br />
* [[Exotic spheres#Brieskorn varieties|Brieskorn spheres]]<br />
* [[Complete intersections]]<br />
* [[Elliptic surfaces]]<br />
* [[Exotic spheres]]<br />
* [[Graph manifolds]]<br />
* [[Hyperbolic manifolds]]<br />
* [[K3 surface]] - there is a [[Wikipedia:K3_surface|Wikipedia page]]<br />
* [[Lens spaces]]<br />
* [[Lie groups]] - we have [[Lie_groups_I:_Definition_and_examples]]<br />
* [[Projective space]] - there is a [[Wikipedia:K3_surface|Wikipedia page]]<br />
**[[Real projective space]]<br />
**[[Complex projective space]]<br />
**[[Quaternionic projective space]]<br />
**[[The Cayley plane]] - there is a short [[Wikipedia:Cayley_plane|Wikipedia page]] and a great [http://mathoverflow.net/questions/1922/what-is-the-cayley-projective-plane MathOverflow page] <br />
* [[Tori]] - there is a [[Wikipedia:Torus|Wikipedia page]]<br />
* [[Seifert fibre spaces]]<br />
* [[Handle-bodies]]<br />
* [[Thickenings]]<br />
<br />
=== Problems ===<br />
<br />
=== Theory ===<br />
* [[Blow up]]<br />
* [[Parametric connected sum#Connected sum|Connected sum]]<br />
* [[Handlebody]]<br />
* [[Plumbing]]<br />
* [[Quotient space]]<br />
<br />
=== History ===</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Manifold_Atlas:Community_PortalManifold Atlas:Community Portal2011-04-01T09:49:12Z<p>Martin Olbermann: /* Manifolds */</p>
<hr />
<div>{{DISPLAYTITLE:{{NAMESPACE}}: {{PAGENAME}}}}<br />
Welcome to the Manifold Atlas Community Portal. Below you will find a list of planned pages on the Atlas - please feel free to get involved and start working on these pages or another page you would like to create.<br />
<br />
Here are some pages you may like to visit:<br />
<br />
* [[Manifold Atlas:Projects|Projects]] you can join.<br />
<br />
* [[Manifold Atlas:Diary|Atlas diary]] for recent events as well as the history of the Atlas.<br />
<br />
* [[Manifold Atlas:The Atlas Observer|Atlas Observer]] lets you sign up for email up-dates about changes at the Atlas.<br />
<br />
* [[Manifold Atlas:Workshops|Workshops]] gives information about Atlas writing workshops.<br />
<br />
* [[Manifold Atlas:Links|Links]] for links to resources about manifolds on the Web.<br />
<br />
== Proposed pages ==<br />
In the first phase of the creation of the Manifold Atlas, we are looking for a broad range of pages. <br />
<br />
* Below are lists of pages we would like to see on the Manifold Atlas. <br />
<br />
* Please feel free to start a page from one of the lists and to add to the discussion page.<br />
<!-- == Pages under construction ==<br />
<br />
A [{{fullurl:Special:WhatLinksHere}}?target=Template:Stub&namespace=0 list] of pages <br />
which contain the [[Template:Stub|“Stub”]] template. --><br />
<br />
* For comparison here is a list of [[Manifold Atlas:Wikipedia pages|Wikipedia pages related to manifolds]].<br />
<br />
=== Manifolds ===<br />
* [[Abelian varieties]]<br />
* [[Arithmetic 3-manifolds]]<br />
* [[Exotic spheres#Brieskorn varieties|Brieskorn spheres]]<br />
* [[Complete intersections]]<br />
* [[Elliptic surfaces]]<br />
* [[Exotic spheres]]<br />
* [[Graph manifolds]]<br />
* [[Hyperbolic manifolds]]<br />
* [[K3 surface]] - there is a [[Wikipedia:K3_surface|Wikipedia page]]<br />
* [[Lens spaces]]<br />
* [[Lie groups]] - we have [[Lie_groups_I:_Definition_and_examples]]<br />
* [[Projective space]] - there is a [[Wikipedia:K3_surface|Wikipedia page]]<br />
**[[Real projective space]]<br />
**[[Complex projective space]]<br />
**[[Quaternionic projective space]]<br />
**[[The Cayley plane]] - there is a short [[Wikipedia:Cayley_plane|Wikipedia page]] and a great [http://mathoverflow.net/questions/1922/what-is-the-cayley-projective-plane MathOverflow page] <br />
* [[Tori]] - there is a [[Wikipedia:Torus|Wikipedia page]]<br />
* [[Seifert fibre spaces]]<br />
* [[Handle-bodies]]<br />
* [[Thickenings]]<br />
<br />
=== Problems ===<br />
<br />
=== Theory ===<br />
* [[Blow up]]<br />
* [[Parametric connected sum#Connected sum|Connected sum]]<br />
* [[Handlebody]]<br />
* [[Plumbing]]<br />
* [[Quotient space]]<br />
<br />
=== History ===</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Manifold_Atlas:Community_PortalManifold Atlas:Community Portal2011-04-01T09:48:30Z<p>Martin Olbermann: /* Manifolds */</p>
<hr />
<div>{{DISPLAYTITLE:{{NAMESPACE}}: {{PAGENAME}}}}<br />
Welcome to the Manifold Atlas Community Portal. Below you will find a list of planned pages on the Atlas - please feel free to get involved and start working on these pages or another page you would like to create.<br />
<br />
Here are some pages you may like to visit:<br />
<br />
* [[Manifold Atlas:Projects|Projects]] you can join.<br />
<br />
* [[Manifold Atlas:Diary|Atlas diary]] for recent events as well as the history of the Atlas.<br />
<br />
* [[Manifold Atlas:The Atlas Observer|Atlas Observer]] lets you sign up for email up-dates about changes at the Atlas.<br />
<br />
* [[Manifold Atlas:Workshops|Workshops]] gives information about Atlas writing workshops.<br />
<br />
* [[Manifold Atlas:Links|Links]] for links to resources about manifolds on the Web.<br />
<br />
== Proposed pages ==<br />
In the first phase of the creation of the Manifold Atlas, we are looking for a broad range of pages. <br />
<br />
* Below are lists of pages we would like to see on the Manifold Atlas. <br />
<br />
* Please feel free to start a page from one of the lists and to add to the discussion page.<br />
<!-- == Pages under construction ==<br />
<br />
A [{{fullurl:Special:WhatLinksHere}}?target=Template:Stub&namespace=0 list] of pages <br />
which contain the [[Template:Stub|“Stub”]] template. --><br />
<br />
* For comparison here is a list of [[Manifold Atlas:Wikipedia pages|Wikipedia pages related to manifolds]].<br />
<br />
=== Manifolds ===<br />
* [[Abelian varieties]]<br />
* [[Arithmetic 3-manifolds]]<br />
* [[Exotic spheres#Brieskorn varieties|Brieskorn spheres]]<br />
* [[Complete intersections]]<br />
* [[Elliptic surfaces]]<br />
* [[Exotic spheres]]<br />
* [[Graph manifolds]]<br />
* [[Hyperbolic manifolds]]<br />
* [[K3 surface]] - there is a [[Wikipedia:K3_surface|Wikipedia page]]<br />
* [[Lens spaces]]<br />
* [[Lie groups]] - we have [[Lie_groups_I:_Definition_and_examples]]<br />
* [[Projective space]] - there is a [[Wikipedia:K3_surface|Wikipedia page]]<br />
**[[Real projective space]]<br />
**[[Complex projective space]]<br />
**[[Quaternionic projective space]]<br />
**[[The Cayley plane]] - there is a short [[Wikipedia:Cayley_plane|Wikipedia page]] and a great [http://mathoverflow.net/questions/1922/what-is-the-cayley-projective-plane|MathOverflow page] <br />
* [[Tori]] - there is a [[Wikipedia:Torus|Wikipedia page]]<br />
* [[Seifert fibre spaces]]<br />
* [[Handle-bodies]]<br />
* [[Thickenings]]<br />
<br />
=== Problems ===<br />
<br />
=== Theory ===<br />
* [[Blow up]]<br />
* [[Parametric connected sum#Connected sum|Connected sum]]<br />
* [[Handlebody]]<br />
* [[Plumbing]]<br />
* [[Quotient space]]<br />
<br />
=== History ===</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Manifold_Atlas:Community_PortalManifold Atlas:Community Portal2011-04-01T09:47:48Z<p>Martin Olbermann: </p>
<hr />
<div>{{DISPLAYTITLE:{{NAMESPACE}}: {{PAGENAME}}}}<br />
Welcome to the Manifold Atlas Community Portal. Below you will find a list of planned pages on the Atlas - please feel free to get involved and start working on these pages or another page you would like to create.<br />
<br />
Here are some pages you may like to visit:<br />
<br />
* [[Manifold Atlas:Projects|Projects]] you can join.<br />
<br />
* [[Manifold Atlas:Diary|Atlas diary]] for recent events as well as the history of the Atlas.<br />
<br />
* [[Manifold Atlas:The Atlas Observer|Atlas Observer]] lets you sign up for email up-dates about changes at the Atlas.<br />
<br />
* [[Manifold Atlas:Workshops|Workshops]] gives information about Atlas writing workshops.<br />
<br />
* [[Manifold Atlas:Links|Links]] for links to resources about manifolds on the Web.<br />
<br />
== Proposed pages ==<br />
In the first phase of the creation of the Manifold Atlas, we are looking for a broad range of pages. <br />
<br />
* Below are lists of pages we would like to see on the Manifold Atlas. <br />
<br />
* Please feel free to start a page from one of the lists and to add to the discussion page.<br />
<!-- == Pages under construction ==<br />
<br />
A [{{fullurl:Special:WhatLinksHere}}?target=Template:Stub&namespace=0 list] of pages <br />
which contain the [[Template:Stub|“Stub”]] template. --><br />
<br />
* For comparison here is a list of [[Manifold Atlas:Wikipedia pages|Wikipedia pages related to manifolds]].<br />
<br />
=== Manifolds ===<br />
* [[Abelian varieties]]<br />
* [[Arithmetic 3-manifolds]]<br />
* [[Exotic spheres#Brieskorn varieties|Brieskorn spheres]]<br />
* [[Complete intersections]]<br />
* [[Elliptic surfaces]]<br />
* [[Exotic spheres]]<br />
* [[Graph manifolds]]<br />
* [[Hyperbolic manifolds]]<br />
* [[K3 surface]] - there is a [[Wikipedia:K3_surface|Wikipedia page]<br />
* [[Lens spaces]]<br />
* [[Lie groups]] - we have [[Lie_groups_I:_Definition_and_examples]]<br />
* [[Projective space]] - there is a [[Wikipedia:K3_surface|Wikipedia page]<br />
**[[Real projective space]]<br />
**[[Complex projective space]]<br />
**[[Quaternionic projective space]]<br />
**[[The Cayley plane]] - there is a short [[Wikipedia:Cayley_plane|Wikipedia page] and a great [http://mathoverflow.net/questions/1922/what-is-the-cayley-projective-plane|MathOverflow page] <br />
* [[Tori]] - there is a [[Wikipedia:Torus|Wikipedia page]<br />
* [[Seifert fibre spaces]]<br />
* [[Handle-bodies]]<br />
* [[Thickenings]]<br />
<br />
=== Problems ===<br />
<br />
=== Theory ===<br />
* [[Blow up]]<br />
* [[Parametric connected sum#Connected sum|Connected sum]]<br />
* [[Handlebody]]<br />
* [[Plumbing]]<br />
* [[Quotient space]]<br />
<br />
=== History ===</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Pin_structuresPin structures2011-04-01T09:34:36Z<p>Martin Olbermann: </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 />
== Introduction ==<br />
<wikitex>;<br />
For an oriented manifold a [[Wikipedia:Spin_structure|spin structure]] is a reduction of the structure group of its <br />
tangent bundle from the connected topological group $SO(n)$ to the double (universal) cover $Spin(n)$.<br />
The non-connected group $O(n)$ has two nontrivial central extensions (double covers) by $\Zz_2$<br />
with different group structures, denoted by $Pin^+(n)$ and $Pin^-(n)$.<br />
A $Pin^+$-structure on a manifold is thus a a reduction of the structure group of its tangent bundle <br />
from $O(n)$ to $Pin^+(n)$, and similarly for $Pin^-$-structures.<br />
<br />
A $Pin^+$-structure on $M$ is equivalent to a $Spin$-structure on $TM\oplus 3 det(TM)$,<br />
a $Pin^-$-structure on $M$ is equivalent to a $Spin$-structure on $TM\oplus det(TM)$.<br />
<br />
The obstruction for existence of a $Pin^+$-structure on $M$ is the characteristic class $w_2(M)$.<br />
If $M$ does admit $Pin$ structures, then the set of isomorphism classes of $Pin^+$-structures on $M$ is acted upon freely<br />
and transitively by $H^1(M;\Zz_2)$.<br />
<br />
The obstruction for existence of a $Pin^-$-structure on $M$ is the characteristic class $w_2(M)+w_1(M)^2$.<br />
If $M$ does admit $Pin$ structures, then the set of isomorphism classes of $Pin^-$-structures on $M$ is acted upon freely <br />
and transitively by $H^1(M;\Zz_2)$.<br />
<br />
A $Pin^\pm$-structure together with an orientation is equivalent to a $Spin$-structure.<br />
<br />
There is also a group $Pin^c$ which is a central extension of $O(n)$ by $S^1$.<br />
$Pin^c$-structures are obstructed by the integral characteristic class $W_3(M)$,<br />
and if they exist, isomorphism classes of $Pin^c$-structures are in bijection with $H^2(M)$.<br />
<br />
For more information on $Pin$-manifolds, including a computation of the low-dimensional bordism groups<br />
of $Pin$-manifolds, see {{cite|Kirby&Taylor1990}}. <br />
</wikitex><br />
<br />
== Examples ==<br />
<wikitex>;<br />
For $k\ge 1$:<br />
<br />
$\RP^{4k}$ admits two $Pin^+$-structures and no $Pin^-$-structure. <br />
<br />
$\RP^{4k+1}$ admits no $Pin^+$-structure and no $Pin^-$-structure. <br />
<br />
$\RP^{4k+2}$ admits no $Pin^+$-structure and two $Pin^-$-structures. <br />
<br />
$\RP^{4k+3}$ admits two $Pin^+$-structures and two $Pin^-$-structures. <br />
</wikitex><br />
<br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Martin Olbermannhttp://www.map.mpim-bonn.mpg.de/Pin_structuresPin structures2011-04-01T09:16:30Z<p>Martin Olbermann: </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 />
== Introduction ==<br />
<wikitex>;<br />
For an oriented manifold a [[Wikipedia:Spin_structure|spin structure]] is a reduction of the structure group of its <br />
tangent bundle from the connected topological group $SO(n)$ to the double (universal) cover $Spin(n)$.<br />
The non-connected group $O(n)$ has two nontrivial central extensions (double covers) by $\Zz_2$<br />
with different group structures, denoted by $Pin^+(n)$ and $Pin^-(n)$.<br />
A $Pin^+$-structure on a manifold is thus a a reduction of the structure group of its tangent bundle <br />
from $O(n)$ to $Pin^+(n)$, and similarly for $Pin^-$-structures.<br />
<br />
A $Pin^+$-structure on $M$ is equivalent to a $Spin$-structure on $TM\oplus 3 det(TM)$,<br />
a $Pin^-$-structure on $M$ is equivalent to a $Spin$-structure on $TM\oplus det(TM)$.<br />
<br />
The obstruction for existence of a $Pin^+$-structure on $M$ is the characteristic class $w_2(M)$.<br />
If $M$ does admit $Pin$ structures, then the set of isomorphism classes of $Pin^+$-structures on $M$ is acted upon freely<br />
and transitively by $H^1(M;\Zz_2)$.<br />
<br />
The obstruction for existence of a $Pin^-$-structure on $M$ is the characteristic class $w_2(M)+w_1(M)^2$.<br />
If $M$ does admit $Pin$ structures, then the set of isomorphism classes of $Pin^-$-structures on $M$ is acted upon freely <br />
and transitively by $H^1(M;\Zz_2)$.<br />
<br />
A $Pin^\pm$-structure together with an orientation is equivalent to a $Spin$-structure.<br />
<br />
For more information on $Pin$-manifolds, including a computation of the low-dimensional bordism groups<br />
of $Pin$-manifolds, see {{cite|Kirby&Taylor1990}}. <br />
</wikitex><br />
<br />
== Examples ==<br />
<wikitex>;<br />
For $k\ge 1$:<br />
<br />
$\RP^{4k}$ admits two $Pin^+$-structures and no $Pin^-$-structure. <br />
<br />
$\RP^{4k+1}$ admits no $Pin^+$-structure and no $Pin^-$-structure. <br />
<br />
$\RP^{4k+2}$ admits no $Pin^+$-structure and two $Pin^-$-structures. <br />
<br />
$\RP^{4k+3}$ admits two $Pin^+$-structures and two $Pin^-$-structures. <br />
</wikitex><br />
<br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Martin Olbermann