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The fundamental groups of presymplectic Hamiltonian G-manifolds

Hui Li

Abstract

We consider presymplectic manifolds equipped with Hamiltonian G-actions, G being a connected compact Lie group. A presymplectic manifold is foliated by the integral submanifolds of the kernel of the presymplectic form. For a presymplectic Hamiltonian G-manifold, Lin and Sjamaar propose a condition under which they show that the moment map image has the same "convex and polyhedral" property as the moment map image of a symplectic Hamiltonian G-manifold, a result proved independently by Atiyah, Guillemin-Sternberg, and Kirwan. In this paper, under the condition Lin and Sjamaar proposed on presymplectic Hamiltonian G-manifolds, we study the fundamental groups of such manifolds, comparing with earlier results on the fundamental groups of symplectic Hamiltonian G-manifolds. We observe that the results on the symplectic case are special cases of the results on the presymplectic case.

The fundamental groups of presymplectic Hamiltonian G-manifolds

Abstract

We consider presymplectic manifolds equipped with Hamiltonian G-actions, G being a connected compact Lie group. A presymplectic manifold is foliated by the integral submanifolds of the kernel of the presymplectic form. For a presymplectic Hamiltonian G-manifold, Lin and Sjamaar propose a condition under which they show that the moment map image has the same "convex and polyhedral" property as the moment map image of a symplectic Hamiltonian G-manifold, a result proved independently by Atiyah, Guillemin-Sternberg, and Kirwan. In this paper, under the condition Lin and Sjamaar proposed on presymplectic Hamiltonian G-manifolds, we study the fundamental groups of such manifolds, comparing with earlier results on the fundamental groups of symplectic Hamiltonian G-manifolds. We observe that the results on the symplectic case are special cases of the results on the presymplectic case.

Paper Structure

This paper contains 10 sections, 27 theorems, 81 equations.

Table of Contents

  1. introduction
  2. clean $G$-actions and moment maps on presymplectic $G$-manifolds
  3. presymplectic $G$-manifolds and the proof of theorems \ref{['O']} and \ref{['A']}
  4. contact $G$-manifolds and the proof of Theorem \ref{['B']}
  5. proof of Theorem \ref{['C']} when the $G$-action is leafwise transitive
  6. the local normal form theorem, the cross section theorem, and a convergence theorem
  7. The fundamental groups of the quotients and the proof of Theorem \ref{['C']}
  8. When $G=T$ is a torus
  9. When $G$ is nonabelian
  10. an application of Theorems \ref{['A']} and \ref{['C']} and two counter examples

Key Result

Theorem 1

Let $(M, \omega)$ be a connected compact presymplectic Hamiltonian $G$-manifold. If the $G$-action is leafwise nontangent everywhere, then the quotient map induces an isomorphism $\pi_1(M)\cong\pi_1(M/G)$.

Theorems & Definitions (54)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1.1
  • Theorem 4
  • Proposition 2.2
  • Theorem 5
  • proof
  • Remark 3.1
  • proof
  • ...and 44 more