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Equivariant Morse theory for Lie algebra actions on Riemannian foliations

Yi Lin, Zuoqin Wang

Abstract

Consider the transverse isometric action of a finite dimensional Lie algebra g on a Riemannian foliation. This paper studies the equivariant Morse-Bott theory on the leaf space of the Riemannian foliations in this setting. Among other things, we establish a foliated version of the Morse-Bott lemma for a g-invariant basic Morse-Bott function, and a foliated version of the usual handle presentation theorem. In the non-equivariant case, we apply these results to present a new proof of the Morse inequalities on Riemannian foliations. In the equivariant case, we apply these results to study Hamiltonian action of an abelian Lie algebra on a presymplectic manifold whose underlying foliation is also Riemannian, and extend the Kirwan surjectivity and injectivity theorem in equivariant symplectic geometry to this situation. Among other things, this implies the Kirwan surjectivity and injectivity hold for Hamiltonian torus actions on symplectic orbifolds.

Equivariant Morse theory for Lie algebra actions on Riemannian foliations

Abstract

Consider the transverse isometric action of a finite dimensional Lie algebra g on a Riemannian foliation. This paper studies the equivariant Morse-Bott theory on the leaf space of the Riemannian foliations in this setting. Among other things, we establish a foliated version of the Morse-Bott lemma for a g-invariant basic Morse-Bott function, and a foliated version of the usual handle presentation theorem. In the non-equivariant case, we apply these results to present a new proof of the Morse inequalities on Riemannian foliations. In the equivariant case, we apply these results to study Hamiltonian action of an abelian Lie algebra on a presymplectic manifold whose underlying foliation is also Riemannian, and extend the Kirwan surjectivity and injectivity theorem in equivariant symplectic geometry to this situation. Among other things, this implies the Kirwan surjectivity and injectivity hold for Hamiltonian torus actions on symplectic orbifolds.
Paper Structure (12 sections, 46 theorems, 76 equations)

This paper contains 12 sections, 46 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. A review of Riemannian foliations
  3. Transverse Lie algebra actions
  4. Transverse action Lie algebroid
  5. Transverse isometric Lie algebra action
  6. Equivariant basic cohomology and Thom isomorphisms
  7. Equivariant Morse theory on Riemannian foliations
  8. Applications to the equivariant basic cohomology theory
  9. Morse inequalities for Riemannian foliations
  10. Hamiltonian actions on pre-symplectic manifolds
  11. Hamiltonian torus actions on symplectic orbifolds
  12. Thom isomorphism for non-orientable foliated vector bundles

Key Result

Proposition 1.1

If the leaf closure space $M/\overline{\mathcal{F}}$ is compact, and if the dimension of $M/\overline{\mathcal{F}}$ is greater than $0$, then there exists a $\mathfrak{g}$-invariant basic function that is Morse-Bott on $(M, \mathcal{F})$. Indeed, the space of $\mathfrak{g}$-invariant basic Morse-Bot

Theorems & Definitions (82)

  • Proposition 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 72 more