The Toral Rank Conjecture and variants of equivariant formality

Amann, Manuel; Zoller, Leopold

doi:10.48550/arxiv.1910.04746

Cited by 5 publications

(8 citation statements)

References 31 publications

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“…It remains to show that P → M is a model for Y T → BT . For this we refer to [7,Proposition,7.17], [2,Proposition 2.7].…”

Section: Lemma 53 With the Setup Above There Is A Mapmentioning

confidence: 99%

“…This turns out to be a more sophisticated problem as is illustrated e.g. by the fact that for 2 ) is realizable by a finite S 1 -CW-complex if and only if c is a square (see Example 6.3). Hence the above realizability question relies heavily on the multiplicative structure, to the point that the outcome might change when passing to R-coefficients.…”

Section: Introductionmentioning

confidence: 99%

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Torus equivariant algebraic models and compact realization

Zoller

2021

Preprint

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Let T be a compact torus. We prove that, up to equivariant rational equivalence, the category of T -simply connected, T -finite type T -spaces with finitely many isotropy types is completely described by certain finite systems of commutative differential graded algebras with consistent choices of degree 2 cohomology classes. We show that the algebraic systems corresponding to finite T -CW-complexes are exactly those which satisfy the necessary condition imposed by the Borel localization theorem along with certain finiteness conditions. We derive an algebraic characterization of when an algebra over a polyonmial ring is realized as the rational equivariant cohomology of a finite T -CW-complex. As further applications we prove that any GKM graph cohomology is realized by a finite T -CW-complex and classify equivariant cohomology algebras of finite S 1 -CW-complexes with discrete fixed points.

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“…It remains to show that P → M is a model for Y T → BT . For this we refer to [7,Proposition,7.17], [2,Proposition 2.7].…”

Section: Lemma 53 With the Setup Above There Is A Mapmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Torus equivariant algebraic models and compact realization

Zoller

2021

Preprint

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“…Recall our definition of two-stage spaces in the introduction which clearly constitutes a slight generalisation of pureness. Two-stage spaces gain special importance due to the following Proposition 1.1 (see Proposition 5.10, p. 32, in [3], cf. [10]).…”

Section: Structure Of the Articlementioning

confidence: 99%

“…the fibration to be π-trivial as well, (and then decompose it as depicted). This follows from the proof of [3,Proposition 5.10] in which we decomposed a two-stage model of X with stage one mapping to a regular sequence in the algebra generated by stage 0 in this form. Without restriction, we may choose the model we start with to be minimal.…”

Section: Structure Of the Articlementioning

confidence: 99%

Homology versus homotopy in fibrations and in limits

Amann

2020

Preprint

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Motivated by prominent problems like the Hilali conjecture Yamaguchi-Yokura recently proposed certain estimates on the relations of the dimensions of rational homotopy and rational cohomology groups of fibre, base and total spaces in a fibration of rationally elliptic spaces.In this article we prove these estimates in the category of formal elliptic spaces and, in general, whenever the total space in addition has positive Euler characteristic or has the rational homotopy type of a homogeneous manifold (respectively of a known example) of positive sectional curvature. Additionally, we provide general estimates approximating the conjectured ones.Moreover, we suggest to study families of rationally elliptic spaces under certain asymptotics, and we discuss the conjectured estimates from this perspective for two-stage spaces.

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“…T (X) and may additionally contain certain Massey products involving elements of p U . This phenomenon is discussed in [3] under the name of spherical actions. Theorem 3.8.…”

Section: Orbit Type Stratificationmentioning

confidence: 99%

Reconstructing the orbit type stratification of a torus action from its equivariant cohomology

Goertsches¹,

Zoller²

2020

Preprint

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We investigate what information on the orbit type stratification of a torus action on a compact space is contained in its rational equivariant cohomology algebra. Regarding the (labelled) poset structure of the stratification we show that equivariant cohomology encodes the subposet of ramified elements. For equivariantly formal actions, we also examine what cohomological information of the stratification is encoded. In the smooth setting we show that under certain conditions -which in particular hold for a compact orientable manifold with discrete fixed point set -the equivariant cohomologies of the strata are encoded in the equivariant cohomology of the manifold.

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The Toral Rank Conjecture and variants of equivariant formality

Cited by 5 publications

References 31 publications

Torus equivariant algebraic models and compact realization

Torus equivariant algebraic models and compact realization

Homology versus homotopy in fibrations and in limits

Reconstructing the orbit type stratification of a torus action from its equivariant cohomology

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