Ro(g)-graded equivariant cohomology theory and sheaves

Abstract

If G is a nite group and if X is a G-space, then a Bredon RO(G)-graded equivariantcohomology theory is dened on X. Furthermore, if X is a G-manifold, thereexists a natural ?ech hypercohomology theory on X. While Bredon RO(G)-gradedcohomology is important in the theoretical aspects, the ?ech cohomology is indispensablewhen computing the cohomology groups. The purpose of this dissertation is toconstruct an isomorphism between these two types of cohomology theories so that theinterplay becomes deeper between the theory and concretely computing cohomologygroups of classical objects. Also, with the aid of ?ech cohomology, we can naturallyextend the Bredon cohomology to the more generalized Deligne cohomology.In order to construct such isomorphism, on one hand, we give a new constructionof Bredon RO(G)-graded equivariant cohomology theory from the sheaf-theoreticviewpoint. On the other hand, with Illman's theorem of smooth G-triangulation ofa G-manifold, we extend the existence of good covers from the nonequivariant tothe equivariant case. It follows that, associated to an equivariant good cover of aG-manifold X, there is a bounded spectral sequence converging to ?ech hypercohomologywhose E1 page is isomorphic to the E1 page of a Segal spectral sequence whichconverges to the Bredon RO(G)-graded equivariant cohomology. Furthermore, Thisisomorphism is compatible with the structure maps in the two spectral sequences. So there is an induced isomorphism between two limiting objects, which are exactly the?ech hypercohomology and the Bredon RO(G)-graded equivariant cohomology.We also apply the above results to real varieties and obtain a quasi-isomorphismbetween two commonly used complexes of presheaves.