# 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 $\ge$${{Stub}} == Introduction == ; 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.) 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 -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} . == 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". === 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 M is a topological manifold. The smoothing problem is addressed by the following {{beginthm|Proposition}}\cite[Proposition 4.2]{Hambleton&Su} Assume \xi is nontrivial. If k is odd, then M admits a smooth structure; if k is even, then M admits a smooth structure if and only if the Kirby-Siebenmann invariant of X is 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$$\Zz_2$, which is the simplest nontrivial group since it has the least number of elements. (Of course if we take the point of view of generators and relations the rank 1 free group $\Zz$$\Zz$ is the simplest one.)

Any other point one should take into account concerning the classification of manifolds with nontrivial fundamental groups $\pi_1$$\pi_1$ is that the higher homotopy groups $\pi_i$$\pi_i$ ($i \ge 2$$i \ge 2$) are modules over the group ring $\Zz[\pi_1]$$\Zz[\pi_1]$, which are apparently homotopy invariants. Especially when we consider $5$$5$-manifolds $M^5$$M^5$, the $\Zz[\pi_1]$$\Zz[\pi_1]$-module structure of $\pi_2(M)$$\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$$M^5$ with fundamental group $\pi_1(M)=\Zz_2$$\pi_1(M)=\Zz_2$, $\pi_2(M)$$\pi_2(M)$ torsion free and is a trivial $\Zz[\Zz_2]$$\Zz[\Zz_2]$-module obtained in [Hambleton&Su] .

## 2 Construction and examples

First some examples known from other context.

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

### 2.1 Circle bundles over simply-connected 4-manifolds

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

Proposition 2.1.[Hambleton&Su, Proposition 4.2]

Assume $\xi$$\xi$ is nontrivial. If $k$$k$ is odd, then
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$M$ admits a smooth structure; if $k$$k$ is even, then
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$M$ admits a smooth structure if and only if the Kirby-Siebenmann invariant of $X$$X$ is $0$$0$.
• $k=1$$k=1$,
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$M$ is a simply-connected 5-manifold. The identification of
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$M$ with manifolds in the standard list of simply-connected 5-manifolds given by the Smale-Barden classification was done in [Duan&Liang].
• $k=2$$k=2$, we have a class of orientable 5-manifolds with fundamental group $\Zz_2$$\Zz_2$, $\pi_2(M)$$\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].

### 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$$\Zz_2$, the connected sum operation is not closed. The connected sum along $S^1$$S^1$" operation $\sharp_{S^1}$$\sharp_{S^1}$ will do the job.

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

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

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## 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$$\Zz_2$, which is the simplest nontrivial group since it has the least number of elements. (Of course if we take the point of view of generators and relations the rank 1 free group $\Zz$$\Zz$ is the simplest one.) Any other point one should take into account concerning the classification of manifolds with nontrivial fundamental groups $\pi_1$$\pi_1$ is that the higher homotopy groups $\pi_i$$\pi_i$ ($i \ge 2$$i \ge 2$) are modules over the group ring $\Zz[\pi_1]$$\Zz[\pi_1]$, which are apparently homotopy invariants. Especially when we consider $5$$5$-manifolds $M^5$$M^5$, the $\Zz[\pi_1]$$\Zz[\pi_1]$-module structure of $\pi_2(M)$$\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$$M^5$ with fundamental group $\pi_1(M)=\Zz_2$$\pi_1(M)=\Zz_2$, $\pi_2(M)$$\pi_2(M)$ torsion free and is a trivial $\Zz[\Zz_2]$$\Zz[\Zz_2]$-module obtained in [Hambleton&Su] . ## 2 Construction and examples First some examples known from other context. • $S^2 \times \mathbb R P^3$$S^2 \times \mathbb R P^3$; • $X^5(q)$$X^5(q)$ $q=1,3,5,7$$q=1,3,5,7$ are the 5-dimensional fake real projective spaces with $X(1)=\mathbb R \mathrm P^5$$X(1)=\mathbb R \mathrm P^5$ (There are exactly 4 in the smooth category). The meaning of $q$$q$ will be clear in the section Invariants". ### 2.1 Circle bundles over simply-connected 4-manifolds Let $X^4$$X^4$ be a closed simply-connected topological 4-manifold, $\xi$$\xi$ be a complex line bundle over $X$$X$ with Chern class $c_1(\xi) \in H^2(X;\Zz)$$c_1(\xi) \in H^2(X;\Zz)$. Let the divisibility of $c_1(\xi)$$c_1(\xi)$ be $k$$k$ (i.e. $c_1(\xi)$$c_1(\xi)$ is $k$$k$ multiple of a primitive element in $H^2(X;\Zz)$$H^2(X;\Zz)$, then the sphere bundle $S(\xi) = \colon M$$S(\xi) = \colon M$ is a 5-manifold with fundament group $\Zz_k$$\Zz_k$ and $\pi_2(M)$$\pi_2(M)$ is a free abelian group of rank $\mathrm{rank} H_2(X)-1$$\mathrm{rank} H_2(X)-1$, and $\pi_2(M)$$\pi_2(M)$ is a trivial module over the group ring. A priori Tex syntax error $M$ is a topological manifold. The smoothing problem is addressed by the following Proposition 2.1.[Hambleton&Su, Proposition 4.2] Assume $\xi$$\xi$ is nontrivial. If $k$$k$ is odd, then Tex syntax error $M$ admits a smooth structure; if $k$$k$ is even, then Tex syntax error $M$ admits a smooth structure if and only if the Kirby-Siebenmann invariant of $X$$X$ is $0$$0$. • $k=1$$k=1$, Tex syntax error $M$ is a simply-connected 5-manifold. The identification of Tex syntax error $M$ with manifolds in the standard list of simply-connected 5-manifolds given by the Smale-Barden classification was done in [Duan&Liang]. • $k=2$$k=2$, we have a class of orientable 5-manifolds with fundamental group $\Zz_2$$\Zz_2$, $\pi_2(M)$$\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]. ### 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$$\Zz_2$, the connected sum operation is not closed. The connected sum along $S^1$$S^1$" operation $\sharp_{S^1}$$\sharp_{S^1}$ will do the job.

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

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

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