5-manifolds with fundamental group of order 2

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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 \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.)

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 [Hambleton&Su] .


[edit] 2 Construction and examples

First some examples known from other context.

  • S^2 \times \mathbb R P^3;
  • X^5(q) q=1,3,5,7 are the 5-dimensional fake real projective spaces with X(1)=\mathbb R \mathrm P^5 (There are exactly 4 in the smooth category). The meaning of q will be clear in the section ``Invariants".


[edit] 2.1 Circle bundles over simply-connected 4-manifolds

Let X^4 be a closed simply-connected topological 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. A priori
Tex syntax error
is a topological manifold. The smoothing problem is addressed by the following

Proposition 2.1.[Hambleton&Su, Proposition 4.2]

Assume \xi is nontrivial. If k is odd, then
Tex syntax error
admits a smooth structure; if k is even, then
Tex syntax error
admits a smooth structure if and only if the Kirby-Siebenmann invariant of X is 0.
  • k=1,
    Tex syntax error
    is a simply-connected 5-manifold. The identification of
    Tex syntax error
    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 orientable 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 [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 \Zz_2, the connected sum operation is not closed. The ``connected sum along S^1" operation \sharp_{S^1} will do the job.

Definition 2.2. Let M_1^5, M_2^5 be two oriented 5-manifolds with \pi_1=\Zz_2. Let E_i \subset M_i be the normal bundle of an embedded S^1 in M_i representing the nontrivial element in the fundamental group. E_i is a rank 4 trivial vector bundle over S^1. Choose trivialisations of E_1 and E_2, and identify the disk bundles of E_1 and E_2 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 M_1 \sharp_{S^1} M_2.

Notice that \sharp_{S^1} is not well-defined, the ambiguity comes from the identification of the two normal bundles, measured by \pi_1SO(4) = \Zz_2. To eliminate the ambiguity we need more structures on the tangent bundles of M_i and require that \sharp_{S^1} 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

$. {{endthm}} * $k=1$, $M$ 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 orientable 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}. === 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 $\Zz_2$, the connected sum operation is not closed. The ``connected sum along $S^1$" operation $\sharp_{S^1}$ will do the job. {{beginthm|Definition}} Let $M_1^5$, $M_2^5$ be two oriented 5-manifolds with $\pi_1=\Zz_2$. Let $E_i \subset M_i$ be the normal bundle of an embedded $S^1$ in $M_i$ representing the nontrivial element in the fundamental group. $E_i$ is a rank 4 trivial vector bundle over $S^1$. Choose trivialisations of $E_1$ and $E_2$, and identify the disk bundles of $E_1$ and $E_2$ 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 $M_1 \sharp_{S^1} M_2$. {{endthm}} Notice that $\sharp_{S^1}$ is not well-defined, the ambiguity comes from the identification of the two normal bundles, measured by $\pi_1SO(4) = \Zz_2$. To eliminate the ambiguity we need more structures on the tangent bundles of $M_i$ and require that $\sharp_{S^1}$ 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. == Invariants == ; ... == Classification/Characterization == ; ... == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]] [[Catefory:5-Manifolds]]\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.)

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 [Hambleton&Su] .


[edit] 2 Construction and examples

First some examples known from other context.

  • S^2 \times \mathbb R P^3;
  • X^5(q) q=1,3,5,7 are the 5-dimensional fake real projective spaces with X(1)=\mathbb R \mathrm P^5 (There are exactly 4 in the smooth category). The meaning of q will be clear in the section ``Invariants".


[edit] 2.1 Circle bundles over simply-connected 4-manifolds

Let X^4 be a closed simply-connected topological 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. A priori
Tex syntax error
is a topological manifold. The smoothing problem is addressed by the following

Proposition 2.1.[Hambleton&Su, Proposition 4.2]

Assume \xi is nontrivial. If k is odd, then
Tex syntax error
admits a smooth structure; if k is even, then
Tex syntax error
admits a smooth structure if and only if the Kirby-Siebenmann invariant of X is 0.
  • k=1,
    Tex syntax error
    is a simply-connected 5-manifold. The identification of
    Tex syntax error
    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 orientable 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 [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 \Zz_2, the connected sum operation is not closed. The ``connected sum along S^1" operation \sharp_{S^1} will do the job.

Definition 2.2. Let M_1^5, M_2^5 be two oriented 5-manifolds with \pi_1=\Zz_2. Let E_i \subset M_i be the normal bundle of an embedded S^1 in M_i representing the nontrivial element in the fundamental group. E_i is a rank 4 trivial vector bundle over S^1. Choose trivialisations of E_1 and E_2, and identify the disk bundles of E_1 and E_2 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 M_1 \sharp_{S^1} M_2.

Notice that \sharp_{S^1} is not well-defined, the ambiguity comes from the identification of the two normal bundles, measured by \pi_1SO(4) = \Zz_2. To eliminate the ambiguity we need more structures on the tangent bundles of M_i and require that \sharp_{S^1} 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

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