5-manifolds with fundamental group of order 2
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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 to the classification of 5-manifolds is to fix a fundamental in advance and consider the classification of manifolds with the given fundamental group. | 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 to 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 $\ | + | 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 $5$-manifolds $M^5$, the $\Zz[\pi_1]$-module structure of $\pi_2(M)$ will play an important role in the classification. | + | 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 \cite{Hambleton&Su} . | ||
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Revision as of 13:00, 3 June 2012
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This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
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 
4. Therefore, a practical approach to 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 
, 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 
is the simplest one.)
Anyother point one should take into account concerning the classification of manifolds with nontrivial fundamental groups 
is that the higher homotopy groups 
(
) are modules over the group ring ![\Zz[\pi_1]](/images/math/c/b/b/cbbd81afbce9307ff44a102af615c005.png)
, which are apparently homotopy invariants. Especially when we consider 
-manifolds 
, the -module structure of 
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 with fundamental group 
, torsion free and is a trivial ![\Zz[\Zz_2]](/images/math/0/6/e/06e45b437bee7ce0e2224a3e03e368b4.png)
-module obtained in [Hambleton&Su] .
2 Construction and examples
...
3 Invariants
...
4 Classification/Characterization
...
5 Further discussion
...
