Free loop spaces and dihedral homology

TL;DR

This paper establishes an $O(2)$-equivariant version of the Jones isomorphism linking free loop space cohomology to dihedral homology, with applications to polynomial forms and computations for the 2-sphere.

Contribution

It introduces an $O(2)$-equivariant Jones isomorphism and extends the de Rham isomorphism to this setting, providing new computational tools.

Findings

Proved an $O(2)$-equivariant Jones isomorphism.

Connected free loop space cohomology with dihedral homology.

Performed explicit computations for the 2-sphere.

Abstract

We prove an $O(2)$-equivariant version of the Jones isomorphism relating the Borel $O(2)$-equivariant cohomology of the free loop space to the dihedral homology of the cochain algebra. We discuss polynomial forms and a variation of the de Rham isomorphism and use these to do a computation for the 2-sphere.