# 5-manifolds with fundamental group of order 2

## 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$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\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 $M$$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 $M$$M$ admits a smooth structure; if $k$$k$ is even, then $M$$M$ admits a smooth structure if and only if the Kirby-Siebenmann invariant of $X$$X$ is $0$$0$.

• $k=1$$k=1$, $M$$M$ is a simply-connected 5-manifold. The identification of $M$$M$ with manifolds in the standard list of simply-connected 5-manifolds given by the Smale-Barden classification was done in [Duan&Liang].
• $k=2$$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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