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

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.

### MSC:

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