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