By Orlando E. Villamayor

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Amer. 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 !

40 III. EQUIVARIANT RATIONAL HOMOTOPY THEORY As a nal comment we mention that the theory of equivariant minimal models has been used by my collaborators and myself to obtain aplications of a more geometric nature, like the classi cation of a large class of G-manifolds up to nite ambiguity and the equivariant formality of G-Kahler manifolds. M. Rothenberg and G. Trianta llou. On the classi cation of -manifolds up to nite ambiguity. Comm. in Pure and Appl. Math. 1991. B. Fine and G. Trianta llou.

Our preferred de nition of homotopy limits is precisely dual. We have a cosimplicial space C (T; D ; S ), the two-sided cobar construction. Its set of n-cosimplices is the product over all f `2 Bn (D ) of the spaces T (d0) S (dn ), topologized as a subspace of Map(Bn(D ); T (d) S (d0)). The f th coordinates of the cofaces and codegeneracies with target Cn(T; D ; S ) are obtained by projecting onto the coordinate of their source that is indexed by the corresponding face or degeneracy applied to f , except that, for the zeroth and last coface, we must compose with T (f1) id : T (d0) S (dn ) ?!

### Algebra Lineal OEA 5 by Orlando E. Villamayor

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