  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.
