Torus actions of complexity one in non-general position

Ayzenberg, Anton; Cherepanov, Vladislav

doi:10.48550/arxiv.1905.04761

Cited by 4 publications

(6 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…From the homological point of view, however, this is the only restriction which we can obtain, at least for the actions of complexity one. In the joint work [5] of the first author and Cherepanov, the following statement was proved. Proposition 3.4 ([5, Thm.2]).…”

Section: General Actions In J-general Positionmentioning

confidence: 87%

See 1 more Smart Citation

Orbit spaces of equivariantly formal torus actions

Ayzenberg¹,

Masuda²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: General Actions In J-general Positionmentioning

confidence: 87%

“…Isomorphism (4) is proved in Lemma 5.9. Equality (5) is the definition of the cosheaf H ˚, see Construction 5.8.…”

Section: A Criterion Of Equivariant Formality In Complexity One Gener...mentioning

confidence: 99%

Orbit spaces of equivariantly formal torus actions

Ayzenberg¹,

Masuda²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…We want to point that our results may have application in the study of positive complexity torus actions which has been recently extensively developing, see for example [1], [3], [14], [26]. In that context the theory of spherical manifolds including theory of homogeneous spaces of compact Lie groups provides numerous examples of positive complexity torus actions, [3].…”

Section: The Main Resultsmentioning

confidence: 96%

“…If W σ ⊂ M 12 then the stratum W σ is defined by the condition P 1i = 0, P 2j = 0 and P pq = 0, for some 3 ≤ i, j ≤ n , 3 ≤ p < q ≤ n. In the local coordinates of the chart M 12 this condition translates to w i = z j = 0 and z p w q = z q w p . Therefore, according to Lemma 14.10 from [8] any stratum W σ ⊂ M 12 is obtained by restricting the surfaces (14) to some C J , where J ⊂ {(3, 3), (3,4), . .…”

Section: An Arbitrary Stratummentioning

confidence: 99%

A resolution of singularities for the orbit spaces $G_{n,2}/T^n$

Buchstaber¹,

Terzić²

2020

Preprint

View full text Add to dashboard Cite

The problem of the description of the orbit space X n = G n,2 /T n for the standard action of the torus T n on a complex Grassmann manifold G n,2 is widely known and it appears in diversity of mathematical questions. A point x ∈ X n is said to be a critical point if the stabilizer of its corresponding orbit is nontrivial. In this paper, the notion of singular points of X n is introduced which opened the new approach to this problem. It is showed that for n > 4 the set of critical points CritX n belongs to our set of singular points SingX n , while the case n = 4 is somewhat special for which SingX 4 ⊂ CritX 4 , but there are critical points which are not singular.The central result of this paper is the construction of the smooth manifold U n with corners, dim U n = dim X n and an explicit description of the projection p n : U n → X n which in the defined sense resolve all singular points of the space X n . Thus, we obtain the description of the orbit space G n,2 /T n combinatorial structure. Moreover, the T n -action on G n,2 is a seminal example of complexity (n − 3) -action. Our results demonstrate the method for general description of orbit spaces for torus actions of positive complexity. 1

show abstract

“…(3) We fix an integer j, and call the action j-independent, if every j tangent weights, at every fixed point x P X T are linearly independent. In the previous works [3,4] we used a different terminology: actions with this property were called actions in j-general position. With these properties satisfied, there holds Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

How is a graph not like a manifold?

Ayzenberg¹,

Masuda²,

Solomadin³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

For an equivariantly formal action of T k on a smooth manifold X 2n with isolated fixed points we investigate the global homological properties of the graded poset S X of face submanifolds. We prove that the condition of j-independency of tangent weights at each fixed point implies pj `1q-acyclicity of the skeleta pS X q r for r ą j `1. This result provides a necessary topological condition for a GKM-graph to be a GKM-graph of some GKM-manifold. We use particular acyclicity arguments to describe the equivariant cohomology algebra of an equivariantly formal manifold X 2n with an pn ´1q-independent action of T n´1 , for n ě 5. This description relates the equivariant cohomology algebra to the face algebra of a certain simplicial poset. This observation relates actions of complexity one to the theory of torus manifolds.

show abstract

Torus actions of complexity one in non-general position

Cited by 4 publications

References 13 publications

Orbit spaces of equivariantly formal torus actions

Orbit spaces of equivariantly formal torus actions

A resolution of singularities for the orbit spaces $G_{n,2}/T^n$

How is a graph not like a manifold?

Contact Info

Product

Resources

About