Surjectivity for Hamiltonian loop group spaces. (English) Zbl 1067.53067

Bott, Raoul; Tolman, Susan; Weitsman, Jonathan

Invent. Math. 155, No. 2, 225-251 (2004).

Let \(G\) be a compact Lie group, and let \(LG\) denote the corresponding loop group. Let \((X,\omega)\) be a weakly symplectic Banach manifold. The authors study a Hamiltonian action of \(LG\) on \((X,\omega)\) with the proper moment map \(\mu:M\rightarrow Lg^{\ast}\). They consider the function \(| \mu| ^2\), and use a version of Morse theory to show that the inclusion map \(j:\mu^{-1}(0)\rightarrow X \) induces a surjection \(j^{\ast}:H_G^{\ast}(X)\rightarrow H_G^{\ast}(\mu^{-1}(0))\), in analogy with Kirwan’s surjectivity theorem in the finite-dimensional case. They also prove a version of this surjectivity theorem for quasi-Hamiltonian \(G\)-spaces.

Reviewer: Wojciech Domitrz (Warszawa)

MSC:

53D20	Momentum maps; symplectic reduction
55N91	Equivariant homology and cohomology in algebraic topology
37J15	Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)

Keywords:

symplectic reduction; momentum map; Morse theory; equivariant cohomology