Math. Z. 182, 485-500 (1983) rvlathematische Zeitschrift 9 Springer-Verlag 1983 Rationalization of Hopf G-Spaces Georgia Vignji6 Triantafillou* Department of Mathematics, University of Chicago, Chicago, Illinois 60637, USA (present address: Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA) w1. Introduction It is well known that the rationalization X o of an H-space X is homotopy equivalent to a product of Eilenberg-Mac Lane spaces, X o ~ @K(rc.(X)| n) n ([4]). The object of this paper is to study the question of whether there is an equivariant splitting of X o in the case that X has a finite group action compatible with the H-structure. Let G be a finite group. Definition 1.1. A Hop/" G-space is a Hopf space (= H-space) on which G acts in such a way that the multiplication m: X x X~ X is an equivariant map, the unit e~X is a fixed point, and the composite X v X_~X x X-+X is G-homo- topic to the folding map V: X v X ~ X. Examples. Let X be a topological group and let G be a finite subgroup of the group of inner automorphisms of X. Then X is obviously a Hopf G-space (the multiplication is a G-map). Another example of a Hopf G-space is the loop space ~2(Y,y) of a G-space Y, where G leaves y fixed. Also Eilenberg-Mac Lane G-spaces are Hopf G-spaces. We recall that an Eilenberg-Mac Lane G-space of dimension n is a G-space K such that the fixed point set K" is an Eilenberg-Mac Lane space of dimension n for every subgroup H of G. Eilenberg-Mac Lane G-spaces are introduced in [13 and are constructed in a way similar to the non-equivariant case. They play the same role in equivariant Bredon cohomology as Eilenberg- Mac Lane spaces in ordinary cohomology. In particular, H"~(X; M)= IX, K(M, n)]G, where M is a coefficient system for G (definition in w K(M, n) is the corresponding Eilenberg-Mac Lane G-space of dimension n, and [, ]~ means G-homotopy classes of G-maps. The equivariant cohomology in turn is the right cohomology in which to express equivariant obstructions for extending or lifting G-maps. * This work was partially supported by a grant from the National Science Foundation.