Bounded Homotopy Equivalences of Hilbert Cube Manifolds

Abstract

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-simplicial complex #^4(Rm X F), the controlled Whitehead space. Using results about approximate fibrations, #"^(Rm 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.

Neighborhoods in stratified spaces with two strata

“…Teardrop neighborhoods can also be used in conjunction with the geometric theory of manifold approximate fibrations [22], [24] to study the geometric topology of manifold stratified pairs. We include two examples here, both of which involve extending a structure on the lower stratum to a neighborhood of the stratum.…”

A Bass–Heller–Swan formula for pseudoisotopies

“…We write P b (X ×R) for the subgroup of bounded, over R, pseudoisotopies [18,20]. There is an isomorphism [1]:…”

“…There is a connection between the Whitehead space and stable pseudoisotopy space given by [1,[6][7][8]18,20]:…”

Control and relaxation over the circle

“…K 1 (C n (Z 1 (X S 1 ))) ( e K 1?n denotes the Whitehead group for n = 0, the reduced K 0 -group if n = 1 and the lower K-groups, K 1?n , if n > 1) The map is the torsion of an element in 0 Wh(X S 1 R n ! R n ) and it is an isomorphism ( 27] For the de nition of (d W ) we use an isomorphism : Wh(X R n ! R n ) !…”