A decomposition theorem forh-cobordant smooth simply-connected compact 4-manifolds

Abstract

Theorem. Let M and N be smooth ;h -cobordant compact 1-connected 4-manifolds. There exist decompositions M = M0 [ M1 and N = N0 [ N1 where M0 and N0 are smooth compact contractible 4-manifolds with boundary ; so that (M1;);and (N1;)are dieomorphic. If M and N are closed; then we may further arrange that M1 and N1 are 1-connected. In fact; if W is an h-cobordism connecting M and N; then W can be written W = W0 [I W1 where (W0;M0;N0)is an (often) nontrivial h-cobordism and (W1;M1;N1)is smoothly the product h-cobordism (M1 I;M1 f0 g;M1 f1 g). This theorem lies in the mainstream of combinatorial 4-manifold theory circa 1975, but, oddly, the result was overlooked. It now ts into a rich context (Donaldson theory) and raises new questions when compared with the known examples. In 1988 Akbulut [A] showed that a manifold X homotopy equivalent to a K3-surface could, after blowing up a point, be transformed into a rational surface Q by cutting out and regluing a certain contractible \Mazur" manifold by an involution of its boundary . At the time a rather delicate argument of Fintushel and Stern (relying on the observed presence of the homology sphere 2;3;7 X) showed that certain Donaldson invariants of X (and its blow up X + ) were nontrivial. This distinguished X + from the h-cobordant manifold Q. In retrospect, this was the rst nontrivial example of our theorem. The result was a surprise: the Donaldson invariant, which derives from the second homology, had been localized on a contractible piece. Recently Bizaca and Gompf [BG] have shown that X is, in fact, dieomorphic to K3, allowing a more direct analysis of its Donaldson invariant. (See e.g. [QK] or [KM].) They also nd an innite family of examples