Equivariant cohomology of torus orbifolds

Darby, Alastair; Kuroki, Shintaro; Song, Jongbaek

doi:10.4153/s0008414x20000760

Cited by 7 publications

(4 citation statements)

References 24 publications

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“…Here T is a compact abelian torus. Subsequently, several papers extended their result to 'nice' stratified spaces see for example [GZ01], [HHH05] and [DKS22]. We note that simplicial GKM orbifold complexes contain the categories of manifolds in the first paragraph.…”

Section: Introductionmentioning

confidence: 90%

Integral equivariant $K$-theory and cobordism ring of simplicial GKM orbifold complexes

Koushik¹,

Sarkar²

2023

Preprint

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In this paper, we define 'simplicial GKM orbifold complexes' and study some of their topological properties. First, we introduce the concept of filtration of regular graphs and 'simplicial graph complexes', which have close relations with simplicial GKM orbifold complexes. We discuss the necessary conditions to confirm an invariant q-cell structure on a simplicial GKM orbifold complex. We introduce 'buildable' and 'divisive' simplicial GKM orbifold complexes. We show that a buildable simplicial GKM orbifold complex is equivariantly formal with rational coefficients, and the integral cohomology of a divisive simplicial GKM orbifold complex has no torsions and is concentrated in even degrees. We give a combinatorial description of the integral equivariant cohomology ring of certain simplicial GKM orbifold complexes. We prove the Thom isomorphism theorem for orbifold G-vector bundles for equivariant cohomology and equivariant K-theory with rational coefficients. We extend the main result of Harada-Henriques-Holm (2005) to the category of G-spaces equipped with 'singular invariant stratification'. We compute the integral equivariant cohomology ring, equivariant K-theory ring and equivariant cobordism ring of divisive simplicial GKM orbifold complexes. We describe the calculation of the integral generalized equivariant cohomology of a divisive simplicial GKM orbifold complex which is not a GKM orbifold.− → H * (X; Q).If ι * is a surjective map then X is called equivariantly formal space. One can also define integrally equivariantly formal space X if the above condition is true with integer coefficients. Note that if H odd (X; Z) = 0 and H * (X; Z) is torsion free then X is integrally equivariantly formal. The readers are referred to [May96] for details and results on G-equivariant cohomology theory H * G . Let X be a compact G-space. The set of all isomorphism classes of complex G-vector bundles on X is an abelian semigroup under the direct sum. Let K G (X) be the associated Grothendieck group. The tensor product of G-vector bundles induces a multiplication structure on K G (X). Then K G (X) is a commutative ring with a unit. For example if X is a point {pt}, then K G (pt) = R(G), the representation ring of G, see [Hus94]. For a locally compact G-space X consider a compact space X + with a base point by the following. If X is not compact then X + is the one-point compactification. If X is already compact then). Thus one can obtain a cohomology theory of Eilenberg-Steenrod [ES52] and known as equivariant K-theory and denoted by Kwhere z denotes the Bott periodicity element having cohomological dimension −2.The G-equivariant ring M U * G (X) is known as equivariant complex cobordism ring, see [tD70]. Sinha [Sin01] and Hanke [Han05] have shown several developments on M U * G .

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Section: Introductionmentioning

confidence: 90%

Integral equivariant $K$-theory and cobordism ring of simplicial GKM orbifold complexes

Koushik¹,

Sarkar²

2023

Preprint

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“…So Γ is obtained by considering the fixed points of the action as vertices and the spindles give the edges if they contain two fixed points. Then one can find the associated axial function α i , connection θ i , c e,e ′ and r e for each GKM orbofold K i , see for example [DKS,Section 2]. This collective data gives (Γ, α, θ).…”

Section: Simplicial Gkm Orbifold Complexes and Its Generalized Cohomo...mentioning

confidence: 99%

Integral generalized equivariant cohomologies of weighted Grassmann orbifolds

Koushik¹,

Sarkar²

2021

Preprint

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In this paper, we define 'simplicial GKM orbifold complexes' and 'simplicial GKM graph complexes. We study some of their topological and combinatorial properties. We discuss some necessary conditions to confirm the invariant q-cell structure on a simplicial GKM orbifold complex. We prove Thom isomrphism for general orbifold Gbundles for equivariant cohomology and equivariant K-theory with rational coefficients. We use this to extend the main result of Harada-Henriques-Holm (2005) to the category of G-spaces equipped with 'singular invariant stratification'. Then we introduce an equivalent definition of weighted Grassmann orbifolds as a generalization of Grassmann manifolds which were discussed by Corti and Reid (2002) and explicitly studied by Abe and Matsumura (2015) to compute their equivariant cohomology. We study their different q-cell structures and discuss some necessary condition to determine if they have invariant CW-structures. We employ the techniques from the first part to study some generalized cohomology theories of weighted Grassmann orbifolds. We introduce 'divisive' weighted Grassmann orbifolds and compute their equivariant cohomology, equivariant K-theory and equivariant cobordism ring with integer coefficients. We discuss Schubert calculus for the equivariant cohomology ring and show how to compute the corresponding structure constants with integral coefficients.

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“…We note that the method in Proposition 4.2 applies to CW-complexes in which are not necessarily manifolds. We also refer to [8, § 5] for the computation of the cohomology ring of toric orbifolds considered in corollary 4.3.…”

Section: The Homotopy Theory Of Complexes Inmentioning

confidence: 99%

The homotopy classification of four-dimensional toric orbifolds

So¹,

Song²

2021

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Let X be a 4-dimensional toric orbifold. If $H^{3}(X)$ has a non-trivial odd primary torsion, then we show that X is homotopy equivalent to the wedge of a Moore space and a CW-complex. As a corollary, given two 4-dimensional toric orbifolds having no 2-torsion in the cohomology, we prove that they have the same homotopy type if and only their integral cohomology rings are isomorphic.

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Equivariant cohomology of torus orbifolds

Abstract: We calculate the integral equivariant cohomology, in terms of generators and relations, of locally standard torus orbifolds whose odd degree ordinary cohomology vanishes. We begin by studying GKM-orbifolds, which are more general, before specialising to half-dimensional torus actions.

Cited by 7 publications

References 24 publications

Integral equivariant $K$-theory and cobordism ring of simplicial GKM orbifold complexes

Integral equivariant $K$-theory and cobordism ring of simplicial GKM orbifold complexes

Integral generalized equivariant cohomologies of weighted Grassmann orbifolds

The homotopy classification of four-dimensional toric orbifolds

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