5-manifolds with fundamental group of order 2

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1 Introduction

The classification of simply-connected 5-manifolds was achieved by Smale [Smale] and Barden [Barden] in the 1960s. The surgery and modified surgery theories provide a toolkit to attack the classification problem of high dimensional manifolds. Dimension 5 is the lowest dimension where the theories apply (except for the 4-dimensional TOP-category), therefore an understanding of the classification of 5-manifolds beyond the Smale-Barden results is expectable. On the other hand, it's known ([Markov], [Kervaire]) that every finitely generated group can be realized as the fundamental group of a manifold of dimension ${{Stub}} == Introduction == <wikitex>; The classification of simply-connected 5-manifolds was achieved by Smale \cite{Smale} and Barden \cite{Barden} in the 1960s. The surgery and modified surgery theories provide a toolkit to attack the classification problem of high dimensional manifolds. Dimension 5 is the lowest dimension where the theories apply (except for the 4-dimensional TOP-category), therefore an understanding of the classification of 5-manifolds beyond the Smale-Barden results is expectable. On the other hand, it's known (\cite{Markov}, \cite{Kervaire}) that every finitely generated group can be realized as the fundamental group of a manifold of dimension $\ge$ 4. Therefore, a practical approach towards the classification of 5-manifolds is to fix a fundamental in advance and consider the classification of manifolds with the given fundamental group. From this point of view, the fist step one might take is the group $\Zz_2$, which is the simplest nontrivial group since it has the least number of elements. (Of course if we take the point of view of generators and relations the rank 1 free group $\Zz$ is the simplest one.) Anyother point one should take into account concerning the classification of manifolds with nontrivial fundamental groups $\pi_1$ is that the higher homotopy groups $\pi_i$ ($i \ge 2$) are modules over the group ring $\Zz[\pi_1]$, which are apparently homotopy invariants. Especially when we consider $-manifolds $M^5$, the $\Zz[\pi_1]$-module structure of $\pi_2(M)$ will play an important role in the classification. And it's not a surprise that the first clear classification result obtained is the trivial module case. Most part of this item will be a survey of the classification result of 5-manifolds $M^5$ with fundamental group $\pi_1(M)=\Zz_2$, $\pi_2(M)$ torsion free and is a trivial $\Zz[\Zz_2]$-module obtained in \cite{Hambleton&Su} . </wikitex> == Construction and examples == === Circle bundles over simply-connected 4-manifolds === <wikitex>; Let $X^4$ be a simply-connected 4-manifold, $\xi$ be a complex line bundle over $X$ with Chern class $c_1(\xi) \in H^2(X;\Zz)$. Let the divisibility of $c_1(\xi)$ be $k$ (i.e. $c_1(\xi)$ is $k$ multiple of a primitive element in $H^2(X;\Zz)$, then the sphere bundle $S(\xi) = \colon M$ is a 5-manifold with fundament group $\Zz_k$ and $\pi_2(M)$ is a free abelian group of rank $\mathrm{rank} H_2(X)-1$, and $\pi_2(M)$ is a trivial module over the group ring. * $k=1$, $M$ is a simply-connected 5-manifold. The identification of $M$ with manifolds in the standard list of simply-connected 5-manifolds given by the Smale-Barden classification was done in \cite{Duan&Liang}. * $k=2$, we have a class of 5-manifolds with fundamental group $\Zz_2$, $\pi_2(M)$ a free abelian group, and a trivial module over the group ring. The classification of these manifolds was the motivation of \cite{Hambleton&Su}. </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex>  == References == {{#RefList:}}   [[Category:Manifolds]] [[Catefory:5-Manifolds]]\ge$ 4. Therefore, a practical approach towards the classification of 5-manifolds is to fix a fundamental in advance and consider the classification of manifolds with the given fundamental group.

From this point of view, the fist step one might take is the group

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$\Zz_2$ , which is the simplest nontrivial group since it has the least number of elements. (Of course if we take the point of view of generators and relations the rank 1 free group $\Zz$ $\Zz$ is the simplest one.)

Anyother point one should take into account concerning the classification of manifolds with nontrivial fundamental groups $\pi_1$ is that the higher homotopy groups $\pi_i$ ( $i \ge 2$ ) are modules over the group ring $\Zz[\pi_1]$ , which are apparently homotopy invariants. Especially when we consider $5$ -manifolds $M^5$ , the $\Zz[\pi_1]$ -module structure of $\pi_2(M)$ will play an important role in the classification. And it's not a surprise that the first clear classification result obtained is the trivial module case.

Most part of this item will be a survey of the classification result of 5-manifolds $M^5$ with fundamental group $\pi_1(M)=\Zz_2$ , $\pi_2(M)$ torsion free and is a trivial $\Zz[\Zz_2]$ -module obtained in [Hambleton&Su] .

2 Construction and examples

2.1 Circle bundles over simply-connected 4-manifolds

Let $X^4$ be a simply-connected 4-manifold, $\xi$ be a complex line bundle over $X$ with Chern class $c_1(\xi) \in H^2(X;\Zz)$ . Let the divisibility of $c_1(\xi)$ be $k$ (i.e. $c_1(\xi)$ is $k$ multiple of a primitive element in $H^2(X;\Zz)$ , then the sphere bundle $S(\xi) = \colon M$ is a 5-manifold with fundament group $\Zz_k$ and $\pi_2(M)$ is a free abelian group of rank $\mathrm{rank} H_2(X)-1$ , and $\pi_2(M)$ is a trivial module over the group ring.

$k=1$ , $M$ is a simply-connected 5-manifold. The identification of $M$ with manifolds in the standard list of simply-connected 5-manifolds given by the Smale-Barden classification was done in [Duan&Liang].
$k=2$ , we have a class of 5-manifolds with fundamental group
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, $\pi_2(M)$ a free abelian group, and a trivial module over the group ring. The classification of these manifolds was the motivation of [Hambleton&Su].

3 Invariants

...

4 Classification/Characterization

...

5 Further discussion

...

6 References

5-manifolds with fundamental group of order 2

Revision as of 13:32, 3 June 2012

Contents

1 Introduction

2 Construction and examples

2.1 Circle bundles over simply-connected 4-manifolds

3 Invariants

4 Classification/Characterization

5 Further discussion

6 References

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@@ Line 13: / Line 13: @@
 == Construction and examples ==
+=== Circle bundles over simply-connected 4-manifolds ===
 <wikitex>;
-...
+Let $X^4$ be a simply-connected 4-manifold, $\xi$ be a complex line bundle over $X$ with Chern class $c_1(\xi) \in H^2(X;\Zz)$. Let the divisibility of $c_1(\xi)$ be $k$ (i.e. $c_1(\xi)$ is $k$ multiple of a primitive element in $H^2(X;\Zz)$, then the sphere bundle $S(\xi) = \colon M$ is a 5-manifold with fundament group $\Zz_k$ and $\pi_2(M)$ is a free abelian group of rank $\mathrm{rank} H_2(X)-1$, and $\pi_2(M)$ is a trivial module over the group ring.
+* $k=1$, $M$ is a simply-connected 5-manifold. The identification of $M$ with manifolds in the standard list of simply-connected 5-manifolds given by the Smale-Barden classification was done in \cite{Duan&Liang}.
+* $k=2$, we have a class of 5-manifolds with fundamental group $\Zz_2$, $\pi_2(M)$ a free abelian group, and  a trivial module over the group ring. The classification of these manifolds was the motivation of \cite{Hambleton&Su}.
 </wikitex>