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 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-manifolds. The first is:

  • \CP^2

The second is a large class of manifolds associated to certain algebraic data. Let \pi be a finitely presentable group. Then for each element \alpha in H_4(K(\pi,1)) there is a smooth, closed, connected, oriented, non-spinnable manifold M(\alpha) with signature zero, fundamental group \pi and 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, 2 and 3: see Oriented bordism to show that there is a closed, smooth, oriented manifold M together with a map f: N \to K(\pi,1) with f_*([M]) = \alpha and signature zero. Then by surgeries on 0- and 1-dimensional spheres one changes M and f in such a way, that M is connected and f_* is an isomorphism on \pi_1 (reference). Finally we form the connected sum with \mathbb{CP}^2 \oplus (-\mathbb {CP}^2) to make sure that 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)

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 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.

5 Realization of the invariants

Given the Theorem above one wonders about the realization of the invariants. Partial answers are easy, but in general this is a complicated open question, where the answer is only known for special fundamental groups. In the simply connected case there are only two invariants, the Euler characteristic and the signature.

The different stable diffeomorphism classes of manifolds with fundamental group \pi are given by 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) }

6 Further discussion


7 References

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