Smooth toral actions

Duflot, Jeanne

doi:10.1016/0040-9383(83)90012-5

Cited by 23 publications

(17 citation statements)

References 2 publications

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“…The two vertical arrows on the left are isomorphisms (one because of excision, and the other is the inverse of the Thom isomorphism), and because D − B κ i has no T κ i -fixed vectors, where T κ i is the isotropy group of B κ i , multiplication with the Euler classes is injective by [13,Proposition 4]. Thus, H * T (M κ+ε , M κ+ε ) → H * T (M κ+ε ) is injective and we obtain short exact sequences…”

Section: Morse-bott Functionsmentioning

confidence: 99%

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Torus actions whose equivariant cohomology is Cohen-Macaulay

Goertsches

Toeben

2010

Journal of Topology

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We study Cohen–Macaulay actions, a class of torus actions on manifolds, possibly without fixed points, which generalizes and has analogous properties as equivariantly formal actions. Their equivariant cohomology algebras are computable in the sense that a Chang–Skjelbred Lemma, and its stronger version, the exactness of an Atiyah–Bredon sequence, hold. The main difference is that the fixed‐point set is replaced by the union of lowest dimensional orbits. We find sufficient conditions for the Cohen–Macaulay property such as the existence of an invariant Morse–Bott function whose critical set is the union of lowest dimensional orbits, or open‐face‐acyclicity of the orbit space. Specializing to the case of torus manifolds, that is, 2r‐dimensional orientable compact manifolds acted on by r‐dimensional tori, the latter is similar to a result of Masuda and Panov, and the converse of the result of Bredon that equivariantly formal torus manifolds are open‐face‐acyclic.

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Section: Morse-bott Functionsmentioning

confidence: 99%

“…Proof. This is essentially Proposition 1 of [13], which is an extended version of [2, Proposition 1.6.2] for tori instead of finite groups. Proof.…”

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confidence: 99%

Torus actions whose equivariant cohomology is Cohen-Macaulay

Goertsches

Toeben

2010

Journal of Topology

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“…The proof uses standard techniques in equivariant cohomology based on work of Quillen [30] and a paper of Duflot [14].…”

Section: Proposition 03 For Any Non-trivial Finite Group the Cohommentioning

confidence: 99%

“…The proof relies heavily on a paper of Duflot [14]. Her work is also used by Henn, Lannes, and Schwartz [23] to prove a result with a similar flavour to regularity.…”

Section: The Elementary Abelian Casementioning

confidence: 99%

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On the Castelnuovo-Mumford regularity of the cohomology ring of a group

Symonds

2010

J. Amer. Math. Soc.

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We record here one easily stated consequence. Proposition 0.3. For any non-trivial finite group, the cohomology ring His generated by elements in degree at most |G| − 1 and the relations between them (as a graded commutative algebra) are generated in degrees at most 2(|G| − 1). This bound is weak, although it can be improved somewhat at the cost of a more complicated formulation, but previously no such bound was known.We go on to prove some other conjectures of Benson on the regularity of the cohomology of other classes of groups [3], specifically for compact Lie groups and virtual Poincaré duality groups.The proof uses standard techniques in equivariant cohomology based on work of Quillen [30] and a paper of Duflot [14].David Green and Simon King have verified Benson's Conjecture computationally for all groups of order less than 256 [20].There is a survey of the properties of group cohomology from the point of view of commutative algebra by Benson [3]. Castelnuovo-Mumford regularityWe will work in the category of Z-graded rings and modules, so maps will respect the grading, elements will be homogeneous, etc., without specific mention. For a module M = i∈Z M i , we will write M ≥d = i≥d M i and similarly for other

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Cohomology of Groups and Unstable Modules over the Steenrod Algebra

Henn

2001

Homotopy Theoretic Methods in Group Cohomology

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Smooth toral actions

Cited by 23 publications

References 2 publications

Torus actions whose equivariant cohomology is Cohen-Macaulay

Torus actions whose equivariant cohomology is Cohen-Macaulay

On the Castelnuovo-Mumford regularity of the cohomology ring of a group

Cohomology of Groups and Unstable Modules over the Steenrod Algebra

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