Construction of right inverses of the Kirwan map

TL;DR

This paper constructs explicit right inverses of the Kirwan map for $S^1$-Hamiltonian spaces using biinvariant diagonal classes and extends this notion to gain additional topological insights, also discussing their non-uniqueness.

Contribution

It introduces a method to explicitly construct right inverses of the Kirwan map via biinvariant diagonal classes for $S^1$-Hamiltonian spaces and extends the concept to a cohomology map.

Findings

Explicit construction of right inverses using multivalued perturbations

Extension of biinvariant diagonal classes to a cohomology map

Proof of non-uniqueness of global biinvariant diagonal classes

Abstract

Biinvariant diagonal classes give rise to right inverses of the Kirwan map. By means of multivalued perturbations of the gradient flow equation such classes are constructed explicitly for $S^1$-Hamiltonian spaces. Moreover, the notion of biinvariant diagonal class is extended to a cohomology map that gives extra topological information on the Hamiltonian space. The non-uniqueness of global biinvariant diagonal classes is also proven.