Equivariant Morse theory for Lie algebra actions on Riemannian foliations

Abstract

We consider a transverse isometric action of a finite-dimensional Lie algebra $\mathfrak g$ on a Riemannian foliation. In this setting, we study equivariant Morse--Bott theory on the leaf space of the foliation. Among other results, we establish a foliated Morse--Bott lemma for $\mathfrak g$-invariant basic Morse--Bott functions and a foliated analogue of the usual handle presentation theorem. In the non-equivariant case, we use these results to give a new proof of the Morse inequalities for Riemannian foliations. In the equivariant case, we apply them to Hamiltonian actions of abelian Lie algebras on presymplectic manifolds whose underlying foliations are Riemannian, and we extend the Kirwan surjectivity and injectivity theorems from equivariant symplectic geometry to this setting. As a consequence, Kirwan surjectivity and injectivity hold for Hamiltonian torus actions on symplectic orbifolds.