5-manifolds with fundamental group of order 2
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== Introduction == | == Introduction == | ||
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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 towards 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 towards the classification of 5-manifolds is to fix a fundamental group in advance and consider the classification of manifolds with the given fundamental group. |
From this point of view, the fist step one might take is the group $\Zz_2$, which is the simplest nontrivial group since it has the least number of elements. (Of course if we take the point of view of generators and relations the rank 1 free group $\Zz$ is the simplest one.) | From this point of view, the fist step one might take is the group $\Zz_2$, which is the simplest nontrivial group since it has the least number of elements. (Of course if we take the point of view of generators and relations the rank 1 free group $\Zz$ is the simplest one.) | ||
− | + | Any other point one should take into account concerning the classification of manifolds with nontrivial fundamental groups $\pi_1$ is that the higher homotopy groups $\pi_i$ ($i \ge 2$) are modules over the group ring $\Zz[\pi_1]$, which are apparently homotopy invariants. Especially when we consider $5$-manifolds $M^5$, the $\Zz[\pi_1]$-module structure of $\pi_2(M)$ will play an important role in the classification. And it's not a surprise that the first clear classification result obtained is the trivial module case. | |
Most part of this item will be a survey of the classification result of 5-manifolds $M^5$ with fundamental group $\pi_1(M)=\Zz_2$, $\pi_2(M)$ torsion free and is a trivial $\Zz[\Zz_2]$-module obtained in \cite{Hambleton&Su} . | 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} . |
Latest revision as of 02:36, 7 February 2013
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Contents |
[edit] 1 Introduction
The classification of simply-connected 5-manifolds was achieved by Smale [Smale] and Barden [Barden] in the 1960s. The surgery and modified surgery theories provide a toolkit to attack the classification problem of high dimensional manifolds. Dimension 5 is the lowest dimension where the theories apply (except for the 4-dimensional TOP-category), therefore an understanding of the classification of 5-manifolds beyond the Smale-Barden results is expectable. On the other hand, it's known ([Markov], [Kervaire]) that every finitely generated group can be realized as the fundamental group of a manifold of dimension 4. Therefore, a practical approach towards the classification of 5-manifolds is to fix a fundamental group in advance and consider the classification of manifolds with the given fundamental group.
From this point of view, the fist step one might take is the group , 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.)
Any other 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] .
[edit] 2 Construction and examples
First some examples known from other context.
- ;
- are the 5-dimensional fake real projective spaces with (There are exactly 4 in the smooth category). The meaning of will be clear in the section ``Invariants".
[edit] 2.1 Circle bundles over simply-connected 4-manifolds
Let be a closed simply-connected topological 4-manifold, be a complex line bundle over with Chern class . Let the divisibility of be (i.e. is multiple of a primitive element in , then the sphere bundle is a 5-manifold with fundament group and is a free abelian group of rank , and is a trivial module over the group ring. A priori is a topological manifold. The smoothing problem is addressed by the following
Proposition 2.1.[Hambleton&Su, Proposition 4.2] Assume is nontrivial. If is odd, then admits a smooth structure; if is even, then admits a smooth structure if and only if the Kirby-Siebenmann invariant of is .
- , is a simply-connected 5-manifold. The identification of with manifolds in the standard list of simply-connected 5-manifolds given by the Smale-Barden classification was done in [Duan&Liang].
- , we have a class of orientable 5-manifolds with fundamental group , a free abelian group, and a trivial module over the group ring. The classification of these manifolds was the motivation of [Hambleton&Su].
[edit] 2.2 Connected sum along $S^1$
In the Smale-Barden's list of simply-connected 5-manifolds, manfolds are constructed from simple building blocks by the connected sum operation. In the world of manifolds with fundamental group , the connected sum operation is not closed. The ``connected sum along " operation will do the job.
Definition 2.2. Let , be two oriented 5-manifolds with . Let be the normal bundle of an embedded in representing the nontrivial element in the fundamental group. is a rank 4 trivial vector bundle over . Choose trivialisations of and , and identify the disk bundles of and using the chosen identification (such that the identification is orientation-reversing, with respect to the induced orientations on the disk bundles from the ambient manifolds), we obtain a new manifold denoted by .
Notice that is not well-defined, the ambiguity comes from the identification of the two normal bundles, measured by . To eliminate the ambiguity we need more structures on the tangent bundles of and require that preserves the structures. (Analogous to the connected sum situation where orientations on manifolds are needed.) This will be explained in more detail in the next section.
[edit] 3 Invariants
...
[edit] 4 Classification/Characterization
...
[edit] 5 Further discussion
...
[edit] 6 References
- [Barden] Template:Barden
- [Duan&Liang] Template:Duan&Liang
- [Hambleton&Su] Template:Hambleton&Su
- [Kervaire] Template:Kervaire
- [Markov] Template:Markov
- [Smale] Template:SmaleCatefory:5-Manifolds
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.)
Any other 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] .
[edit] 2 Construction and examples
First some examples known from other context.
- ;
- are the 5-dimensional fake real projective spaces with (There are exactly 4 in the smooth category). The meaning of will be clear in the section ``Invariants".
[edit] 2.1 Circle bundles over simply-connected 4-manifolds
Let be a closed simply-connected topological 4-manifold, be a complex line bundle over with Chern class . Let the divisibility of be (i.e. is multiple of a primitive element in , then the sphere bundle is a 5-manifold with fundament group and is a free abelian group of rank , and is a trivial module over the group ring. A priori is a topological manifold. The smoothing problem is addressed by the following
Proposition 2.1.[Hambleton&Su, Proposition 4.2] Assume is nontrivial. If is odd, then admits a smooth structure; if is even, then admits a smooth structure if and only if the Kirby-Siebenmann invariant of is .
- , is a simply-connected 5-manifold. The identification of with manifolds in the standard list of simply-connected 5-manifolds given by the Smale-Barden classification was done in [Duan&Liang].
- , we have a class of orientable 5-manifolds with fundamental group , a free abelian group, and a trivial module over the group ring. The classification of these manifolds was the motivation of [Hambleton&Su].
[edit] 2.2 Connected sum along $S^1$
In the Smale-Barden's list of simply-connected 5-manifolds, manfolds are constructed from simple building blocks by the connected sum operation. In the world of manifolds with fundamental group , the connected sum operation is not closed. The ``connected sum along " operation will do the job.
Definition 2.2. Let , be two oriented 5-manifolds with . Let be the normal bundle of an embedded in representing the nontrivial element in the fundamental group. is a rank 4 trivial vector bundle over . Choose trivialisations of and , and identify the disk bundles of and using the chosen identification (such that the identification is orientation-reversing, with respect to the induced orientations on the disk bundles from the ambient manifolds), we obtain a new manifold denoted by .
Notice that is not well-defined, the ambiguity comes from the identification of the two normal bundles, measured by . To eliminate the ambiguity we need more structures on the tangent bundles of and require that preserves the structures. (Analogous to the connected sum situation where orientations on manifolds are needed.) This will be explained in more detail in the next section.
[edit] 3 Invariants
...
[edit] 4 Classification/Characterization
...
[edit] 5 Further discussion
...