5-manifolds with fundamental group of order 2
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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 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 , 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 , 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 -module obtained in [Hambleton&Su] .
2 Construction and examples
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3 Invariants
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4 Classification/Characterization
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5 Further discussion
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