[120] Surjectivity for Hamiltonian Loop Group Spaces

Chapter
Part of the Contemporary Mathematicians book series (CM)

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.

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Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  3. 3.Department of MathematicsUniversity of CaliforniaSanta CruzUSA

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