[120] Surjectivity for Hamiltonian Loop Group Spaces

Abstract

Let G be a compact Lie group, and let LG denote the corresponding loop group. Let (X, ω) be a weakly symplectic Banach manifold. Consider a Hamiltonian action of LG on (X, ω), and assume that the moment map $\mu \,:\,\,{\rm X}\,\, \to L{g^ * }$ is proper. We consider the function ${\left| \mu \right|^2}:\,X\, \to \,\mathbb{R}$, and use a version of Morse theory to showthat the inclusion map $j\,:\,{\mu ^{ - 1}}\left( 0 \right)\, \to \,X$ induces a surjection $j{\,^ * }:\,\,H_G^ * \left( X \right)\, \to \,H_G^ * \left( {{\mu ^{ - 1}}\left( 0 \right)} \right)$, in analogywithKirwan’s surjectivity theorem in the finite-dimensional case. We also prove a version of this surjectivity theorem for quasi-Hamiltonian G-spaces.