# Stable classification of 4-manifolds

## 1 Introduction


## 2 Construction and examples

We begin with the construction of manifolds which give many stable diffeomorphism types of $4$$4$-manifolds:

• $S^4$$S^4$
• $S^2 \times S^2$$S^2 \times S^2$
• $\CP^2$$\CP^2$
• $K:= \{x \in \CP^3 | \sum x_i^4 =0\}$$K:= \{x \in \CP^3 | \sum x_i^4 =0\}$, the Kummer surface.

Let $P=$$P=$ be the presentation of a group $\pi$$\pi$. Then we build a $2$$2$-dimensional complex $X(P)$$X(P)$ by taking a wedge of $n$$n$ circles and attaching a $2$$2$-cell via each relation $r_i$$r_i$. Then we thicken $X(P)$$X(P)$ to a smooth compact manifold with boundary $W(P)$$W(P)$ in $\mathbb R^5$$\mathbb R^5$ and consider its boundary denoted by $M(P)$$M(P)$. For details and why this is well defined see Thickenings. $M(P)$$M(P)$ is a smooth $4$$4$-manifold with fundamental group $\pi$$\pi$ and we add it to our list:

• $M(P)$$M(P)$

Let $\pi$$\pi$ be a finitely presentable group. Then for each element $\alpha$$\alpha$ in $H_4(K(\pi_1,1)$$H_4(K(\pi_1,1)$ there is a smooth, closed, oriented manifold $M(\alpha)$$M(\alpha)$ with signature zero and a map $f: M(\alpha) \to K(\pi_1,1)$$f: M(\alpha) \to K(\pi_1,1)$ mapping the fundamental class to $\alpha$$\alpha$. The existence follows (for example using the Atiyah-Hirzebruch spectral sequence B-Bordism#Spectral sequences from the fact that the oriented bordism group is zero in degree $1$$1$, $2$$2$ and $3$$3$ Oriented bordism. This manifold is - of course - not unique. But we will see that its stable diffeomorphism class is unique, if we require that the universal covering is non spinnable. We add it to our list:

• $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/Characterization

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 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)$.

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$ Stable classification of manifolds#The normal k-type. 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 Stable classification of manifolds#Theorem 3.1.

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