×

Pseudo-isotopies, K-theory, and homotopy theory. (English) Zbl 0656.57020

Homotopy theory, Proc. Symp., Durham/Engl. 1985, Lond. Math. Soc. Lect. Note Ser. 117, 35-71 (1987).
[For the entire collection see Zbl 0628.00011.]
From the introduction: “The problem we address in this paper is that of studying the space of pseudo-isotopies of a manifold. This group, which we denote by P(M), is defined to be the group of diffeomorphisms \(P(M)=Diff(M\times I,\quad \partial M\times I\cup M\times \{0\})\) where I denotes the unit interval, \(\partial M\) denotes the boundary of M, and where the above notation refers to diffeomorphisms of \(M\times I\) that equal the identity in a neighbourhood of \(\partial M\times I\cup M\times \{0\}\). In this paper we will give an expository account of certain advances in the past fifteen years in understanding the homotopy type of the pseudo-isotopy space, P(M), using K-theoretic and homotopy theoretic methods.”... “The aim of this paper is to describe in a cohesive manner recent results of many people in this exciting area of research. It is purely expository. The details of the proofs all appear in other papers.” It should be noted, however, that the paper by G. Carlsson, R. Cohen, and W.-C. Hsiang [K-Theory 1, 53-82 (1987; Zbl 0649.55001)] to which the “main theorem” of the paper (Theorem 4 in the introduction) refers, does not exist; the announcement was withdrawn.
Reviewer: F.Waldhausen

MSC:

57R52 Isotopy in differential topology
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
55P15 Classification of homotopy type
55P99 Homotopy theory
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology