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Equivariant basic cohomology of singular Riemannian foliations

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Francisco C. Caramello Jr

Differential Geometry · Submitted 25 Sep 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that, modulo torsion, its equivariant basic cohomology localizes to the set of closed leaves of the foliation, in the spirit of the classical localization theorem of Borel.

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arXiv ID2209.12334 (version 4)
Pith IDNX2PKPK4, Pith's permanent record of this paper and its review
CitationFrancisco C. Caramello Jr. "Equivariant basic cohomology of singular Riemannian foliations." arXiv:2209.12334 (2022). https://pith.science/paper/NX2PKPK4
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,738 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.40 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.76 · compares with prior results 0.58 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; .

We introduce the notion of equivariant basic cohomology for singular Riemannian foliations with transverse infinitesimal actions, and prove some elementary properties such as its invariance under homotopies. For the particular case of singular Killing foliations, there is the natural transverse action of its structural algebra. We prove that, modulo torsion, its equivariant basic cohomology localizes to the set of closed leaves of the foliation, in the spirit of the classical localization theorem of Borel. As applications, we obtain that the basic Euler characteristic also localizes to this set, and that the dimension of the basic cohomology of the localized foliation is less than or equal to that of the whole foliation, with equality occurring precisely in the equivariantly formal case.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We introduce the notion of equivariant basic cohomology for singular Riemannian foliations with transverse infinitesimal actions, and prove some elementary properties such as its invariance under homotopies. contribution 1.00 0.96 0.14 asserted 1.00
s1 For the particular case of singular Killing foliations, there is the natural transverse action of its structural algebra. background 0.89 0.19 0.03 asserted 1.00
s2 We prove that, modulo torsion, its equivariant basic cohomology localizes to the set of closed leaves of the foliation, in the spirit of the classical localization theorem of Borel. result 0.99 0.95 0.66 asserted 1.00
s3 As applications, we obtain that the basic Euler characteristic also localizes to this set, and that the dimension of the basic cohomology of the localized foliation is less than or equal to that of the whole foliation, with equality occurring precisely in the equivariantly formal case. result 0.98 0.95 0.25 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.94 · evidence from proof or derivation 1.00.

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The paper
Original abstract

We introduce the notion of equivariant basic cohomology for singular Riemannian foliations with transverse infinitesimal actions, and prove some elementary properties such as its invariance under homotopies. For the particular case of singular Killing foliations, there is the natural transverse action of its structural algebra. We prove that, modulo torsion, its equivariant basic cohomology localizes to the set of closed leaves of the foliation, in the spirit of the classical localization theorem of Borel. As applications, we obtain that the basic Euler characteristic also localizes to this set, and that the dimension of the basic cohomology of the localized foliation is less than or equal to that of the whole foliation, with equality occurring precisely in the equivariantly formal case.

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