Homotopy equivalences in equivariant topology

Abstract

Homomorphisms up to homotopy (higher homotopies that is) are generalized for the equivariant category. Homotopy equivalences have an inverse in this new category. Introduction. In equivariant topology the notion of a homotopy equivalence presents a problem. Strictly within the equivariant category, homotopy equivalence seems to be too limited a concept,1 e.g. the unit interval I (acting on itself as an H-space) is not of the same equivariant homotopy type as {1} as a {1}-space. Some of the rather general tools used in homotopy theory of topological groups and H-spaces (e.g. studying classifying spaces) can also be used in equivariant homotopy theory. In this paper we use G.-maps between G-spaces (as defined in 1.4, and similar to H.-maps between H-spaces) to study a new notion of homotopy equivalence. Roughly speaking, if X and X are G-spaces and if f: X -> X is a G-equivariant map and also an ordinary homotopy equivalence, then there exists a sequence of maps gn: X x (I x G )nf X forming a G.-map such thatf and the maps {gn} form a pair of G.-homotopy equivalences. The complete theorem is stated in ?2. The proof includes the proof of the corresponding theorem on H-spaces as stated in [2, Theorem 4.1] or in [1] as Proposition 1.17. We hope to convince the reader that this proof is not as messy as it is described by the authors of [1] on p. 13. 1. Definitions. DEFINITION 1.1. An H-space G is a topological space with a continuous multiplication u. We assume that u is strictly associative. No unit element is needed. DEFINITION 1.2. A topological space X is called a G-space, if G acts on X in a continuous and in an associative manner. Multiplication and actions will be denoted by the usual juxtaposition. DEFINITION 1.3. An H. -map h from G to G of length r2 is a sequence of continuous maps hn: G x (Q, x G)n )l G such that Presented to the Society, August 22, 1975; received by the editors August 13, 1974 and, in revised form, November 4, 1975. AMS (MOS) subject classifications (1970). Primary 55D45, 55DlO; Secondary 57E99. i C. N. Lee, A. Wasserman (see [5]), and T. Petrie have relatively simple examples of pairs of manifolds X and Y with S1-actions and an S1-mapf: X -> Y such thatf is an ordinary homotopy equivalence. But there is no equivariant map g: Y -* X which is a homotopy equivalence. 2 We will mention the length of maps only if it is essential to the context. ? American Mathematical Society 1976