9704819 Hsiang One of the most astounding results of Donaldson is as follows. There are h-cobordisms (W; M, N) of simply-connected smooth 4-manifolds M and N such that M and N are not diffeomorphic. They are closed 4-manifolds with the same non-trivial second homology over the reals, and the non-triviality of the h-cobordism is detected by gauge invariants. It was observed by Hsiang et al. Inv. Math. 123 (1996) 343-348 that the cobordism W may be decomposed into two cobordisms, the first being a trivial one and the second being a non-trivial relative h-cobordism of contractible 4-manifolds bounded by the same 3-dimensional homology sphere. This means that all the non-triviality of W is concentrated inside a contractible piece. Because the gauge invariants heavily depend on the non-triviality of the quadratic form of the second homology of M, two questions need to be answered. (A) How can one detect the non-triviality of a cobordism (W; M, N) of contractible 4-manifolds M, N (relative to the boundaries)? In fact, there should be combinatorial invariants related to the classical knot invariants to detect this non-triviality, and, in particular, one should be able to describe some explicit examples constructed from the explicit non-trivial h-cobordisms of Akbulut. (B) How do any such combinatorial invariants relate to gauge theory. Hsiang is seeking solutions to these two problems. He has already found that (B) is related to Property P for 3-manifolds. Geometric topology treats spaces known as manifolds that come in a vast array of configurations and all different dimensions. A manifold of dimension n is referred to briefly as an n-manifold, and what the dimension means is that an ant living on the manifold would think himself in n-dimensional Euclidean space if he took only short excursions. However, he could learn otherwise on returning from a lengthy journey and finding himself disoriented or constrained in some way. Think of an ant on life s upport that requires trailing an umbilicus behind him attached at the other end to a base station. Already in the 2-dimensional case, he will discover if he travels widely enough that there are differences between living on the surface of a sphere and living on the surface of a doughnut, even if the sphere (or doughnut) is very large and appears to his myopic gaze to be everywhere a flat Euclidean plane. The 2-dimensional possibilities have been understood since the 19th century, but the possibilities become increasingly various and impossible to classify fully as the dimension increases. What is always surprising to the unitiate is to learn that in spite of this dimensional complication, some of the most intriguing problems, problems that have been solved for the higher dimensions, remain unsolved for dimensions 3 or 4 or both, a fact of importance to beings who live mostly in these dimensions and wish, for example, to understand the physics of their cosmos. The present project takes note of the exciting new methods that have been introduced into low-dimensional topology in recent years, first by Donaldson, and more recently by Seiberg and Witten, the so-called gauge theoretic methods. Then it asks some incisive questions about how these gauge invariants relate to the vast arsenal of techniques with which algebraic topologists had previously been equipped. ***
