Year: 2026
Type: preprint
Abstract: We prove an equivariant version of the theorem of Hector, Mac\'ıas-Virgós, and Sanmart\'ın-Carbón 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 \[ Ω^\bullet(M/H)\congΩ^\bullet(M,\mathcal F)^H \] 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.
Source: arXiv (Cornell University)
Author: Lin, Yi
Institution: Georgia Southern University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7160305021