Products in Equivariant Homology

Abstract

We refine the intersection product in homology to an equivariant setting, which unifies several known constructions. As an application, we give a common generalisation of the Chas-Sullivan string product on a manifold and the Chataur-Menichi string product on the classifying space by defining a string product on the Borel construction of a manifold. We prove a vanishing result which enables us to define a secondary product. The secondary product is then used to construct secondary versions of the Chataur-Menichi string product, and the equivariant intersection product in the Borel equivariant homology of a manifold with an action of a compact Lie group. The latter reduces to the product in homology of the classifying space defined by Kreck, which coincides with the cup product in negative Tate cohomology if the group is finite.