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

  • Yi Lin
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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 ℱ be the foliation by H0-orbits. We prove that pullback induces an isomorphism Ω•(M/H)≅Ω•(M,ℱ)H provided that H is second countable or that the induced action of H/H0 on M/H0 satisfies a natural subduction condition. We also give a smooth free action for which descent fails, showing that an additional hypothesis is genuinely necessary.

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Cite

BibTeXRIS
BibTeX
@misc{lin2026descent,
  title = {{Descent of Basic Forms to Quotients by Locally Free Lie Group Actions}},
  author = {Yi Lin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.01891},
  eprint = {2605.01891},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.01891},
}
RIS
TY  - UNPB
TI  - Descent of Basic Forms to Quotients by Locally Free Lie Group Actions
AU  - Yi Lin
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2605.01891
UR  - https://arxiv.org/abs/2605.01891
AB  - 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 ℱ be the foliation by H_0-orbits. We prove that pullback induces an isomorphism Ω^•(M/H)≅Ω^•(M,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.
ER  -