Construction of right inverses of the Kirwan map

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.