By an h-cobordism (W,, M) on an n-manifold M we mean an (n + 1)-manifold W,, together with an inclusion M c dW,, with bicollared boundary, such that both M c W and tgW-M c W are proper homotopy equivalences. The most obvious example is the trivial h-cobordism (M • I, M x 0). By an s-cobordism theorem we mean a theorem which gives necessary and sufficient conditions for a smooth, PL or topological h-cobordism to be smoothly, piecewise linearly or topologically equivalent rel M to the trivial h-cobordism, or more generally a calculation of the set of isomorphism classes of h-cobordisms on M in the chosen category. The s-cobordism theorems have played a key role in numerous aspects of geometric topology, including the classification of manifolds by surgery.The first s-cobordism theorem in which nontrivial obstructions appear was given by Barden [Ba], Mazur [Ma] and Stallings [Sta]. They showed that the smooth or PL isomorphism classes of h-cobordisms on a compact manifold M of dimension >5 are in one to one correspondence with elements of the Whitehead group of n~M. Here, the torsion of an h-cobordism (W,M) may be represented geometrically by the spine of a relative handlebody decomposition of(W, M).Siebenmann ] extended this classification to h-cobordisms on non-compact manifolds, with the invariants lying in his infinite simple homotopy groups.Kirby and Siebenmann [KSi, Essay III] showed that if M is a topological manifold, then the same invariants classify the topological isomorphism classes of h-cobordisms on M. Their proof relies on three of their theorems which have had a striking impact on topological manifold theory in general.The first, Concordance Implies Isotopy, states (in strongly relative form) that if h: (M x l, M)~(W,,M) is a homeomorphism onto a smooth or PL hcobordism on M (dim M> 5). then h may be approximated arbitrarily closely by a smooth or PL isomorphism rel M.Second, their Product Structure Theorem states (also in relative form) that the isotopy classes of smooth or PL structures on M • ~-, (dim M > 5) are in * Partially supported by the NSF and by the Graduate School of Northern Illinois University 62 M. Steinberger one to one correspondence with the isotopy classes of such structures on M via crossing with the standard structure on R.Finally, their Handlebody Existence Theorem, which is a direct consequence of the two theorems above, states that if N is a codimension zero submanifold of t~M with bicollared boundary and dim M > 6, then (M, N) has a relative handlebody decomposition. This has been of particular importance in extending certain smooth or PL arguments, such as Siebenmann's End Theorem ISis] to the topological category.We mention these results specifically because our approach to the associated equivariant problem is by identifying the obstructions to these phenomena in the equivariant context.Later, Chapman ECI] , using his theorem that handle straightening is unobstructed in Hilbert cube manifolds, showed that two locally compact complexes are simple homotopy equivalent if ...
