Equivariant Basic Cohomology of Riemannian Foliations

Goertsches, Oliver; Toeben, Dirk

doi:10.48550/arxiv.1004.1043

Cited by 6 publications

(23 citation statements)

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“…, where a ⊥ is a Lie subalgebra of a which acts transversely on F ′ and q a constant which relates to the codimension of F and F ′ . Notice that if F ′ = F , F W degenerates to points and a ⊥ = a, that is [7] Proposition 4.9. Similar to the definition of equivariant basic Chern characters, the equivariant basic Â-genus character of normal bundle Âa ⊥ (νF ) resp.…”

Section: Introductionmentioning

confidence: 92%

“…We denote by s and r for the source and target map on all groupoids. Proposition 2.1 ([14] Proposition 5.6 Example 5.8 (7)). The holonomy groupoid of a foliation has a natural Lie groupoid structure.…”

Section: Preliminariesmentioning

confidence: 99%

“…Let ã be the commuting sheaf of ( M , F ), [7]. This section is devoted to discuss the relation between F ′ and ã.…”

Section: Lifted Foliations and Commuting Sheafmentioning

confidence: 99%

“…Let H • a (M, F ) be the a-equivariant basic de Rham cohomology with polynomial coefficients, see [6,11]. Goertsches and Töben [7] proved a cohomology isomorphism H • a (M, F ) ≃ H • SO(q) (W ). Duflo and Vergne [4] extended the equivariant cohomology to the case of C ∞ -coefficients.…”

Section: Introductionmentioning

confidence: 99%

“…If F ′ = F , the first step will be degenerated, that is Molino's theory. Section 6 is devoted to the generalization of cohomological isomorphism proposed in [7] Proposition 4.9. In this section, the foliation F (resp.…”

Section: Introductionmentioning

confidence: 99%

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A generalization of Molino's theory and equivariant basic Â-genus characters

Liu

2021

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Molino's theory is a mathematical tool for studying Riemannian foliations. In this paper, we propose a generalization of Molino's theory with two Riemannian foliations. For this purpose, the projection of foliation with respect to a fibration is discussed. The generalization results in an equivariant basic cohomological isomorphism in case of Killing foliation. It is a generalization of results given by Goertsches and Töben. We also give a geometric realization of the cohomological isomorphism through equivariant basic Â-genus characters, who play a prominent role in calculating the index of an elliptic operator by Atiyah-Singer's index formula.

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Section: Introductionmentioning

confidence: 92%

Section: Preliminariesmentioning

confidence: 99%

“…Let ã be the commuting sheaf of ( M , F ), [7]. This section is devoted to discuss the relation between F ′ and ã.…”

Section: Lifted Foliations and Commuting Sheafmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A generalization of Molino's theory and equivariant basic Â-genus characters

Liu

2021

Preprint

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Equivariant cohomology of K-contact manifolds

2012

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We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen-Macaulay, the natural substitute of equivariant formality for torus actions without fixed points. As a consequence, generic components of the contact moment map are perfect Morse-Bott functions for the basic cohomology of the orbit foliation F of the Reeb flow. Assuming that the closed Reeb orbits are isolated, we show that the basic cohomology of F vanishes in odd degrees, and that its dimension equals the number of closed Reeb orbits. We characterize K-contact manifolds with minimal number of closed Reeb orbits as real cohomology spheres. We also prove a GKM-type theorem for K-contact manifolds which allows to calculate the equivariant cohomology algebra under the nonisolated GKM condition.2010 Mathematics Subject Classification. 53D35, 53D20, 55N25.

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