SMOOTH NONTRIVIAL ^DIMENSIONAL s-COBORDISMS

This announcement exhibits smooth 4-dimensional manifold triads (W;Mo,Mi) which are s-cobordisms, i.e. the inclusions Mi Ç W, i = 0,1, are simple homotopy equivalences, but are not diffeomorphic or even homeomorphic to a product Mi x [0,1]. The Barden-Mazur-Stallings s-cobordism theorem constitutes one of the foundational stones of modern topology. It asserts, in the smooth, piecewiselinear, or topological categories, that if W is a manifold of dimension at least six, with boundary components M», i = 0,1, whose inclusions into W are simple homotopy equivalences, then W is necessarily a product (see [K, H, RS, KS]). For simply connected smooth manifolds of dimension at least six, this result had already been proven by Smale as the "fo-cobordism theorem" [8m], with the generalized Poincaré conjecture in higher dimensions as a corollary. The s-cobordism statement holds in dimensions one and two, and is equivalent to the Poincaré conjecture in dimension three. Freedman [Fl, F2] proved the five-dimensional result for topological manifolds with fundamental group of polynomial growth (e.g. finite or polycyclic). Donaldson's extraordinary results imply the failure of the five-dimensional result in the smooth (or piecewise linear) category even for simply connected manifolds; by [Fl] the resulting h-cobordisms will still be topological products. Using Freedman's results, the present authors produced some nontrivial orientable four-dimensional topological s-cobordisms [CS1, CS2]. (See [MS] for a nonorientable and definitely nonsmoothable example.) These topological constructions have been further studied and extended by Kwasik and Schultz [KwS]. We will now use a different construction to produce some nontrivial smooth s-cobordisms. Neither the construction nor the proof rely on any of the results cited above. Let M be a quaternionic space-form; i.e. M = Mr = S /Qr, Qr the quaternionic group of order 2 r + 2 . Then it is well known that the orientable manifold M has a one-sided Heegaard splitting