Products in Equivariant Homology

TL;DR

This paper develops an equivariant intersection product in homology, unifying existing string products and introducing secondary products, with applications to manifolds and classifying spaces under group actions.

Contribution

It generalizes the string product to an equivariant setting and constructs secondary products, connecting various known products in homology and cohomology.

Findings

Unified string product on Borel constructions of manifolds.

Defined secondary products extending known string products.

Connected equivariant homology products to Tate cohomology.

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.