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Math. Soc. 274(1982), 509-532. 3. Rational equivariant Hopf spaces In spite of the conceptual analogy of the equivariant theory to the nonequivariant one, the calculations in the equivariant case are much more subtle and can yield surprising results. We illustrate this by describing our work on rational Hopf Gspaces. It is a basic feature of nonequivariant homotopy theory that the rational Hopf spaces split as products of Eilenberg-Mac Lane spaces. The equivariant analogue is false. By a Hopf G-space we mean a based G-space X together with a G-map X X !

Quillen. Rational Homotopy Theory, Ann of Math. 90(1968), 205-295. D. Sullivan. In nitesimal Computations in Topology, Publ. Math. IHES 47(1978), 269-332. 2. Equivariant minimal models For nite groups G an analogous theory can be developed for G-rational homotopy types of G-simplicial complexes. For simplicity we assume throughout that the spaces X are G-connected and G-simply connected, which means that each xed point space X H is connected and simply connected; however, the theory works just as well for G-nilpotent spaces.

14(1982), 231-237. CHAPTER III Equivariant Rational Homotopy Theory by Georgia Trianta llou 1. Summary: the theory of minimal models Let G be a nite group. In this chapter, we summarize our work on the algebraicization of rational G-homotopy theory. To simplify the statements we assume simply connected spaces throughout the chapter. The theory can be extended to the nilpotent case in a straightforward manner. We recall that by rationalizing a space X , we approximate it by a space X0 the homotopy groups of which are equal to (X ) Q, thus neglecting the torsion.