Stable classification of 4-manifolds

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1 Introduction

In this page we report about the stable classification of closed oriented 4-manifolds. We will begin with a special class of closed oriented 4-manifolds, namely those, where the universal covering is not spinnable.

2 Construction and examples

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

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

Let P=<g_1, \dots , g_n| r_1,\dots,r_m> be the presentation of a group \pi. Then we build a 2-dimensional complex X(P) by taking a wedge of n circles and attaching a 2-cell via each relation r_i. Then we thicken X(P) to a smooth compact manifold with boundary W(P) in \mathbb R^5 and consider its boundary denoted by M(P). For details and why this is well defined see Thickenings. M(P) is a smooth 4-manifold with fundamental group \pi and we add it to our list:

  • M(P)

Let \pi be a finitely presentable group. Then for each element \alpha in H_4(K(\pi,1) there is a smooth, closed, oriented non-spinnable manifold M(\alpha) with signature zero, fundamental group \pi and u_*([M]) = \alpha. This is proved by using the Atiyah-Hirzebruch spectral sequence) and the fact that the oriented bordism groups are zero in degree 1, 2 and 3: see Oriented bordism to show that there is a closed, smooth, oriented manifold N together with a map f: N \to (K(\pi,1) with f_*([N]) = \alpha.

and a map f: M(\alpha) \to K(\pi_1,1) mapping the fundamental class to \alpha. The existence follows (for example using the Atiyah-Hirzebruch spectral sequence) from the fact that the oriented bordism groups are zero in degree 1, 2 and 3: see 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)

3 Invariants

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

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

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

4 Classification

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

The different stable diffeomorphism classes of manifolds with fundamental group \pi are given by 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-type is K(\pi,1) \times BSO \to BO, see Stable classification of manfifolds. Thus the B-bordism group is \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) under the signature and the image of the fundamental class. Now the statement follows from Theorem 3.1 of Stable classification of manifolds.

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

5 Further discussion


6 References

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