Families of subgroups and completion

Abstract

For a given compact Lie group G and a family of subgroups 3r of G, a classifying space E ~ is defined. It is a G-space such that every subgroup of G having fixed points on EJbelongs to ~ and such that for every H e F the space E ~ is Hcontractible i.e. it has an H-equivariant contraction onto a point. We want to compute the equivariant cohomology of classifying spaces for families of subgroups. Let h o be an equivariant multiplicative cohomology theory. For any subgroup H e J , we define an ideal I(H)=ker{hG(pt)-~h~(G/H)}. The set of ideals { / ( H I ) . . . I ( H n ) I H i E Y} defines a topology on hc(pt ) which is called the ~topology. The completion conjecture says that for a 'nice' cohomology theory h e and a 'nice' space X the projection X x E~--,X induces an isomorphism/~x(J) • ho(X) --