Orbit spaces of torus actions on Hessenberg varieties

Cherepanov, Vladislav

doi:10.48550/arxiv.1905.02294

Cited by 3 publications

(8 citation statements)

References 12 publications

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“…Is it possible to characterize equivariant formality in terms of the topology of the orbit space? The results of [7] suggested that the answer is negative: the complexity one torus actions on regular semisimple Hessenberg varieties are equivariantly formal, but they have orbit spaces with nontrivial topology.…”

Section: Preliminariesmentioning

confidence: 99%

“…Remark 3.6. In [7], an argument similar to Lemma 3.5 was applied to show that homology in degrees 0,1,2 vanish for the orbit spaces of complexity one torus actions on regular semisimple Hessenberg varieties. We suppose that vanishing of homology in degrees 0,1,2 is a general phenomenon for the orbit spaces of equivariantly formal torus actions of complexity one with isolated fixed points.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…It is easy to prove that, under certain technical assumptions [1], the orbit space Q n`1 " X 2n {T n´1 is a closed topological manifold if the tangent weights are in general position at each fixed point. If at least one fixed point has weights not in general position, then the orbit space Q n`1 is a manifold with boundary [7]. Karshon and Tolman [12] proved that the orbit space of any Hamiltonian torus action of complexity one in general position is homeomorphic to the sphere S n`1 .…”

Section: Introductionmentioning

confidence: 99%

“…We see that in complexity one, general position of tangent weights implies strong topological constraints on the structure of the orbit space. However, the second author studied the complexity one torus actions on regular semisimple Hessenberg varieties in [7]: these actions are not in general position and their orbit spaces have more interesting topology. One example of the orbit space of a Hessenberg variety is the 5-sphere with four discs cut off, and another example is the 6-sphere with a tubular neighbourhood of a graph cut off.…”

Section: Introductionmentioning

confidence: 99%

“…By the discussion above, there exist a large number of actions of complexity one, whose orbit space is homeomorphic to the sphere: they all correspond to weights in general position. If the weights are not in general position, the orbit space is an pn `1q-manifold with boundary, according to [7], so we cannot get H n`1 pQ n`1 q -Z. Therefore, the case L " B∆ n´1 is exceptional in the sense that it corresponds to the general position of weights.…”

Section: Introductionmentioning

confidence: 99%

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Torus actions of complexity one in non-general position

Ayzenberg¹,

Cherepanov²

2019

Preprint

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Let the compact torus T n´1 act on a smooth compact manifold X 2n effectively with nonempty finite set of fixed points. We pose the question: what can be said about the orbit space X 2n {T n´1 if the action is cohomologically equivariantly formal (which essentially means that H odd pX 2n ; Zq " 0). It happens that homology of the orbit space can be arbitrary in degrees 3 and higher. For any finite simplicial complex L we construct an equivariantly formal manifold X 2n such that X 2n {T n´1 is homotopy equivalent to Σ 3 L. The constructed manifold X 2n is the total space of the projective line bundle over the permutohedral variety hence the action on X 2n is Hamiltonian and cohomologically equivariantly formal. We introduce the notion of the action in j-general position and prove that, for any simplicial complex M , there exists an equivariantly formal action of complexity one in j-general position such that its orbit space is homotopy equivalent to Σ j`2 M .

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Section: Preliminariesmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Torus actions of complexity one in non-general position

Ayzenberg¹,

Cherepanov²

2019

Preprint

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Orbit spaces of equivariantly formal torus actions

Ayzenberg¹,

Masuda²

2019

Preprint

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Let a compact torus T " T n´1 act on a smooth compact manifold X " X 2n effectively, with nonempty finite set of fixed points, and suppose that stabilizers of all points are connected. If H odd pXq " 0 and the weights of tangent representation at each fixed point are in general position, we prove that the orbit space Q " X{T is a homology pn`1q-sphere. If, in addition, π 1 pXq " 0, then Q is homeomorphic to S n`1 . We introduce the notion of j-generality of tangent weights of torus action. For any action of T k on X 2n with isolated fixed points and H odd pXq " 0, we prove that j-generality of weights implies pj `1q-acyclicity of the orbit space Q. This statement generalizes several known results for actions of complexity zero and one. In complexity one, we give a criterion of equivariant formality in terms of the orbit space. In this case, we give a formula expressing Betti numbers of a manifold in terms of certain combinatorial structure that sits in the orbit space.

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Toric orbit spaces which are manifolds

Ayzenberg¹,

Gorchakov²

2023

Preprint

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We characterize the actions of compact tori on smooth manifolds for which the orbit space is a topological manifold (either closed or with boundary). For closed manifolds the result was originally proved by Styrt in 2009. We give a new proof for closed manifolds which is also applicable to manifolds with boundary. In our arguments we use the result of Provan and Billera who characterized matroid complexes which are pseudomanifolds. We study the combinatorial structure of torus actions whose orbit spaces are manifolds. In two appendix sections we give an overview of two theories related to our work. The first one is the combinatorial theory of Leontief substitution systems from mathematical economics. The second one is the topological Kaluza-Klein model of Dirac's monopole studied by Atiyah. The aim of these sections is to draw some bridges between disciplines and motivate further studies in toric topology.

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Orbit spaces of torus actions on Hessenberg varieties

Cited by 3 publications

References 12 publications

Torus actions of complexity one in non-general position

Torus actions of complexity one in non-general position

Orbit spaces of equivariantly formal torus actions

Toric orbit spaces which are manifolds

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