# Stable classification of 4-manifolds

## 1 Introduction

In this page we report about the stable classification of closed oriented $4$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}4$-manifolds. We will begin with a special class of closed oriented $4$$4$-manifolds, namely those, where the universal covering is not spinnable.

## 2 Construction and examples I

We begin with the construction of two classes of manifolds which can be used to give many stable diffeomorphism types of non-spinnable $4$$4$-manifolds. The first is:

• $\CP^2$$\CP^2$

The second is a large class of manifolds associated to certain algebraic data. Let $\pi$$\pi$ be a finitely presentable group. Then for each element $\alpha$$\alpha$ in $H_4(K(\pi,1))$$H_4(K(\pi,1))$ there is a smooth, closed, connected, oriented, non-spinnable manifold $M(\alpha)$$M(\alpha)$ with signature zero, fundamental group $\pi$$\pi$ and $u_*([M]) = \alpha$$u_*([M]) = \alpha$. This is proved in several steps by first using the Atiyah-Hirzebruch spectral sequence) and the fact that the oriented bordism groups are zero in degree $1$$1$, $2$$2$ and $3$$3$: see Oriented bordism to show that there is a closed, smooth, oriented manifold $M$$M$ together with a map $f: N \to K(\pi,1)$$f: N \to K(\pi,1)$ with $f_*([M]) = \alpha$$f_*([M]) = \alpha$ and signature zero. Then by surgeries on $0$$0$- and $1$$1$-dimensional spheres one changes $M$$M$ and $f$$f$ in such a way, that $M$$M$ is connected and $f_*$$f_*$ is an isomorphism on $\pi_1$$\pi_1$ (reference). Finally we form the connected sum with $\mathbb{CP}^2 \oplus (-\mathbb {CP}^2)$$\mathbb{CP}^2 \oplus (-\mathbb {CP}^2)$ to make sure that $M$$M$ is non-spinnable. This manifold is of course not unique but we will see that it is unique up to stable diffeomorphisms and we abbreviate it by

• $M(\alpha)$$M(\alpha)$

## 3 Invariants

The following is a complete list of invariants for the stable classification of closed, smooth oriented $4$$4$-manifolds whose universal covering is not spinnable:

• The Euler characteristic $\chi (M)$$\chi (M)$
• The signature $\sigma (M)$$\sigma (M)$
• The fundamanetal group $\pi_1(M)$$\pi_1(M)$
• The image of the fundamental class $[u_*([M])]\in H_4(K(\pi_1(M),1)/Out(\pi_1(M))$$[u_*([M])]\in H_4(K(\pi_1(M),1)/Out(\pi_1(M))$of $M$$M$.

Here $u:M \to K(\pi_1(M),1)$$u:M \to K(\pi_1(M),1)$ is a classifying map of the universal covering and $Out(\pi_1(M))$$Out(\pi_1(M))$ is the outer automorphism group which acts on the homology of $K(\pi_1(M),1)$$K(\pi_1(M),1)$.

## 4 Classification

Theorem 4.1. Let $M$$M$ and $N$$N$ be $4$$4$-dimensional compact smooth manifolds with non-spinnable universal covering. Then $M$$M$ and $N$$N$ are stably diffeomorphic if and only if the invariants above agree.

The proof of this result is an easy consequence of the general stable classification theorem ([Kreck1999], Stable classification of manifolds). Namely, the normal $1$$1$-type is $K(\pi,1) \times BSO \to BO$$K(\pi,1) \times BSO \to BO$, see Stable classification of manfifolds. Thus the $B$$B$-bordism group is $\Omega ^{SO}(K(\pi_1,1)$$\Omega ^{SO}(K(\pi_1,1)$, which by the Atiyah-Hirzebruch spectral sequence is ismorphic to $\mathbb Z \oplus H_4(K(\pi,1);\mathbb Z)$$\mathbb Z \oplus H_4(K(\pi,1);\mathbb Z)$ under the signature and the image of the fundamental class. Now the statement follows from Theorem 3.1 of Stable classification of manifolds.

The different stable diffeomorphism classes of manifolds with fundamental group $\pi$$\pi$ are given by $M(\alpha ) \sharp_k \CP^2 \sharp_s (-\CP^2)$$M(\alpha ) \sharp_k \CP^2 \sharp_s (-\CP^2)$. Here \$k+s + \chi

$\displaystyle \xymatrix{ B \ar[r]^{} \ar[d]^{} & K(\pi, 1) \ar[d]^{\hat w_2} \\ BSO \ar[r]^{w_2} & K(\Zz/2, 2) }$

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