Equivariant de Rham cohomology: theory and applications

Goertsches, Oliver; Zoller, Leopold

doi:10.1007/s40863-019-00129-4

Cited by 10 publications

(12 citation statements)

References 66 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By the comparison theorem [23,Thm. A.22], it suffices to show that 1 ⊗ p * 1 induces an isomorphism on the first pages of the spectral sequences which are S(g * ) ⊗ H * (M) and S(g * ) ⊗ H * (M × EG) [23,Thm. A.8], respectively.…”

Section: Basic Cohomologymentioning

confidence: 99%

“…Let 1 on the cochain level. The latter map respects a filtration which gives rise to a spectral sequence that computes the equivariant cohomology, see [23,Section A.3]). By the comparison theorem [23,Thm.…”

Section: Basic Cohomologymentioning

confidence: 99%

“…The latter map respects a filtration which gives rise to a spectral sequence that computes the equivariant cohomology, see [23,Section A.3]). By the comparison theorem [23,Thm. A.22], it suffices to show that 1 ⊗ p * 1 induces an isomorphism on the first pages of the spectral sequences which are S(g * ) ⊗ H * (M) and S(g * ) ⊗ H * (M × EG) [23,Thm.…”

Section: Basic Cohomologymentioning

confidence: 99%

“…A.8], respectively. Going through the proof of [23,Thm. A.8] shows that this induced map is just the map 1 ⊗ p * 1 .…”

Section: Basic Cohomologymentioning

confidence: 99%

See 3 more Smart Citations

A Boothby–Wang Theorem for Besse Contact Manifolds

Kegel

Lange

2020

Arnold Math J.

View full text Add to dashboard Cite

A Reeb flow on a contact manifold is called Besse if all its orbits are periodic, possibly with different periods. We characterize contact manifolds whose Reeb flows are Besse as principal $$S^1$$ S 1 -orbibundles over integral symplectic orbifolds satisfying some cohomological condition. Apart from the cohomological condition, this statement appears in the work of Boyer and Galicki in the language of Sasakian geometry (Boyer and Galicki in Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008). We illustrate some non-commonly dealt with perspective on orbifolds in a proof of the above result. More precisely, we work with orbifolds as quotients of manifolds by smooth Lie group actions with finite stabilizer groups. By introducing all relevant orbifold notions in this equivariant way, we avoid patching constructions with orbifold charts. As an application, and building on work by Cristofaro-Gardiner–Mazzucchelli, we deduce a complete classification of closed Besse contact 3-manifolds up to strict contactomorphism.

show abstract

Section: Basic Cohomologymentioning

confidence: 99%

Section: Basic Cohomologymentioning

confidence: 99%

Section: Basic Cohomologymentioning

confidence: 99%

“…A.8], respectively. Going through the proof of [23,Thm. A.8] shows that this induced map is just the map 1 ⊗ p * 1 .…”

Section: Basic Cohomologymentioning

confidence: 99%

See 2 more Smart Citations

A Boothby–Wang Theorem for Besse Contact Manifolds

Kegel

Lange

2020

Arnold Math J.

View full text Add to dashboard Cite

show abstract

“…We will see a transverse counterpart of this result below. We refer to [46] and [27] for more detailed introductions to the classical equivariant cohomology theory, and to [9] and [32] for thorough treatments of this topic.…”

Section: Transverse Topology Of Killing Foliationsmentioning

confidence: 99%

Leaf closures of Riemannian foliations: a survey on topological and geometric aspects of Killing foliations

Alexandrino¹,

Caramello²

2020

Preprint

View full text Add to dashboard Cite

A smooth foliation is Riemannian when its leaves are locally equidistant. The closures of the leaves of a Riemannian foliation on a simply connected manifold, or more generally of a Killing foliation, are described by flows of transverse Killing vector fields. This offers significant technical advantages in the study of this class of foliations, which nonetheless includes other important classes, such as those given by the orbits of isometric Lie group actions. Aiming at a broad audience, in this survey we introduce Killing foliations from the very basics, starting with a brief revision of the main objects appearing in this theory, such as pseudogroups, holonomy and basic cohomology. We then review Molino's structural theory for Riemannian foliations and present its transverse counterpart in the theory of complete pseudogroups of isometries, emphasizing the connections between these topics. We also survey some classical results and recent developments in the theory of Killing foliations. Finally, we review some topics in the theory of singular Riemannian foliations and discuss singular Killing foliations, also proposing a new approach to them via holonomy metric pseudogroups and the theory of blow-ups, which possibly opens up a new area of interest. Contents2010 Mathematics Subject Classification. 53C12.

show abstract

Equivariant basic cohomology of singular Riemannian foliations

Caramello

2023

Monatsh Math

View full text Add to dashboard Cite

Equivariant de Rham cohomology: theory and applications

Abstract: This is a survey on the equivariant cohomology of Lie group actions on manifolds, from the point of view of de Rham theory. Emphasis is put on the notion of equivariant formality, as well as on applications to ordinary cohomology and to fixed points.

Cited by 10 publications

References 66 publications

A Boothby–Wang Theorem for Besse Contact Manifolds

A Boothby–Wang Theorem for Besse Contact Manifolds

Leaf closures of Riemannian foliations: a survey on topological and geometric aspects of Killing foliations

Equivariant basic cohomology of singular Riemannian foliations

Contact Info

Product

Resources

About