Geometric transfer and the homotopy type of the automorphism groups of a manifold

Abstract

Introduction. For M a compact manifold with or without boundary in one of the categories Diff, PL or Top we denote by A(M) the semisimplicial (ss) group of automorphisms of M (i.e. the group of diffeomorphisms, PL homeomorphisms or homeomorphisms, resp.). In this paper we describe (in the stability range and localized away from two) the homotopy structure of A(M). We do this in terms of A(M), the ss group of block automorphisms (whose homotopy structure is a sophisticated formulation of surgery theory [ABK]) and some new homotopy functors 5 + (Theorem C). The -functors 5 and 5 'L here considered have already been studied in the PL category by Hatcher [H] from a different point of view. The present geometric description is appropriate to the description of the homotopy structure of A(M) and enables us to produce a natural ,B: SC(X) -* 5 (X) from the associative monoid of simple homotopy equivalence of X to 5 (X) which will be important for our study. Besides the geometric topology results [BLR] two ideas partially developed in [BLI are basic. The first is the transfer map and stability for concordances, described in the PL context by Hatcher and in the differentiable context in [BLI. The second is the splitting of a Z(1) weak commutative (topological) group with involution as a product up to homotopy of two pieces, the symmetric, respectively antisymmetric parts. Applied to our situation those parts manifest very different behavior. The results of this paper have been sketched by the first author in the survey paper [B]. There is an overlap between some (but not all) results of ?A and results of Hatcher [H2], published in the same volume as [B]. The results of this paper are stated in the next section (statement of results) and the proofs are given in subsequent sections. We continue this section with a description of the notations used.