Descent of Basic Forms to Quotients by Locally Free Lie Group Actions

Abstract

We prove an equivariant version of the theorem of Hector, Mac\'{\i}as-Virg\'os, and Sanmart\'{\i}n-Carb\'on identifying diffeological forms on the leaf space of a foliation with basic forms. For any group $K$ acting by foliation-preserving diffeomorphisms, we show that this identification is an isomorphism of $K$-equivariant cochain complexes. We then establish a descent theorem for basic differential forms under locally free Lie group actions without assuming properness. Let a Lie group $H$, not necessarily connected or second countable, act smoothly and locally freely on a second countable manifold $M$, and let $\mathcal F$ be the foliation by $H_0$-orbits. We prove that pullback induces an isomorphism \[ \Omega^\bullet(M/H)\cong\Omega^\bullet(M,\mathcal F)^H \] provided that $H$ is second countable or that the induced action of $H/H_0$ on $M/H_0$ satisfies a natural subduction condition. We also give a smooth free action for which descent fails, showing that an additional hypothesis is genuinely necessary.