The Equivariant Serre Spectral Sequence

For spaces with a group action, we introduce Bredon cohomology with local (or twisted) coefficients and show that it is invariant under weak equivariant homotopy equivalence. We use this new cohomology to construct a Serre spectral sequence for equivariant fibrations. Bredon [1] introduced what is now called Bredon cohomology, with the purpose of developing obstruction theory in the context of spaces equipped with an action of a fixed group G. The kind of coefficients needed in this theory are not abelian groups but rather contravariant functors from the orbit category tf (tr) into abelian groups. The purpose of this paper is to prove the existence of a spectral sequence for a Serre fibration of (7-spaces. As in the nonequivariant case, if no further restrictions are made, it is necessary to use cohomology with twisted coefficients. Such twisted coefficients are not functors on the orbit category cf(G), but on some augmentation of cf (G) depending on the base space of the fibration. Our approach is to give a new definition of Bredon cohomology in terms of cohomology of categories. Indeed, with a G-space X we shall associate a category AG(X) of "equivariant singular simplices in X." Any abelian groupvalued functor Af from the orbit category cf(G) can be viewed as a system of coefficients on the category AG(X). A key result is that for any such Af, the cohomology groups H*(AGX, M) of this category are naturally isomorphic to the Bredon cohomology groups HG(X, M); see Theorem 2.2. The category AG(X) allows us to define cohomology groups of a G-space with more general coefficients. In particular, we shall construct a category HG(X) of "equivariant homotopy classes of paths in X," which sits in between AG(X) and cf (G) by functors AG(X) -► UG(X) -► tf(G). A twisted or local system of coefficients Af on X is then defined as an abelian group-valued functor on HG(X), and the Bredon cohomology of X with twisted coefficients can now be constructed as the cohomology H*(AG(X), M) of the category AG(X). This is invariant under weak G-homotopy equivalence, see Theorem 2.3. (The converse is also true: a map of G-spaces is a weak G-homotopy equivalence whenever it induces an equivalence of fundamental groupoids as well as an Received by the editors September 21, 1990 and, in revised form, August 19, 1991. 1991 Mathematics Subject Classification. Primary 55N91, 55R91, 55T91. The authors gratefully acknowledge financial support from the Swedish Natural Science Research Council and the Netherlands Science Organisation (NWO). ©1993 American Mathematical Society 0002-9939/93 $1.00+ $.25 per page