Bounded homotopy equivalences of Hilbert cube manifolds

Abstract

Let M and F be Hubert cube manifolds with F compact. The purpose of this paper is to study homotopy equivalences/: M -» R' X F which have bounded control in the Redirection. Roughly, these homotopy equivalences form a semi-sim- plicial complex #^4(Rm X F), the controlled Whitehead space. Using results about approximate fibrations, x F ) is related to the semi-simplicial complex of bounded concordances on R' X F. Then the homotopy groups of #^(Rm X F) are computed in terms of the lower algebraic AT-theoretic functors K_i. 1. Introduction. Let F be a compact Hilbert cube manifold. We are interested in homotopy equivalences /: M -» Rm X F which are controlled in the Redirection, where M is also a Hilbert cube manifold. To say / is controlled in the Redirection means that pf: M -» Rm is an approximate fibration, where p: Rm X F -* Rm is projection. The collection of all such homotopy equivalences, which are additionally given to be retractions onto the collared submanifold Rm X F of M, form the vertices of a semi-simplicial complex ?TA(Rm X F) (see §2 for the precise definition). The main result of this paper is the computation of the homotopy groups of WA(Rm X F) (see Corollary 1 below). In order to do this we relate WA(Rm X F) to the semi-simplicial complex ^h(Rm X F) of bounded concordances on Rm X F. An «-simplex of <eb^Rm X F) is a homeomorphism