Manifolds with Tti

In this note we announce some results extending results of S. P. Novikov ([6] and [7]), the author [2], and C. T. C. Wall [8]. In the above papers it is shown how to characterize the homotopy type of 1-connected smooth closed manifolds of dimension w ^ S , « ^ 2 mod 4, and how to reduce the diffeomorphy classification of such manifolds to homotopy theory, with similar results for bounded manifolds in [8], We show how to adapt these techniques to manifolds with TT\ — Z and get analogous results. By studying the "mapping torus" using these results one may obtain results on existence and pseudo-isotopy of diffeomorphisms, (see [3]). One has for example the situation of a closed smooth manifold M and a map ƒ : M—>X, such that the normal bundle v of M in S is induced by ƒ from a bundle £ over X. One does surgery on M with respect to the map ƒ, i.e. if W is the cobordism determined by the surgery, then ƒ extends to a map F: W-+X such that the normal bundle of W in SXl is induced from £ by F. In case M is simply connected many conditions facilitate the surgery, such as the Whitney embedding theorem, and the Hurewicz theorem, so that, with appropriate hypothesis on X and £, it is often possible to do surgery to create a manifold homotopy equivalent to X. The case of a nonzero fundamental group poses many problems, but if itxM — Z, one can reduce the situation to the simply connected case by using extra geometrical structure. The idea is to consider a 1-connected manifold U with two 1-connected boundary components, dU — Ao^JAi, with ƒ: A0—*A\ a diffeomorphism, and consider the identification space M n