GKM theory for orbifold stratified spaces and application to singular toric varieties

Sarkar, Soumen; Song, Jongbaek

doi:10.1016/j.topol.2020.107472

Cited by 6 publications

(4 citation statements)

References 38 publications

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“…If C tPiu consists of simplices, we call (2.4) a ∆-conic sequence, which agrees with the definition of a retraction sequence studied in [BSS17,BNSS21]. A polytope admitting a ∆-conic sequence is referred to as an almost simple polytope in [SS21]. We note that every simple polytope has at least one ∆-conic sequence (see [BSS17, Proposition 2.3]), which is not true for arbitrary convex polytope.…”

Section: Conic Sequence Of a Convex Polytopementioning

confidence: 79%

Conic decomposition of a toric variety and its application to cohomology

Park¹,

Song²

2021

Preprint

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We introduce the notion of a conic sequence of a convex polytope. It is a way of building up a polytope starting from a vertex and attaching faces one by one with certain regulations. We apply this to a toric variety to obtain an iterated cofibration structure on it. This allows us to prove several vanishing results in the rational cohomology of a toric variety and to calculate Poincaré polynomials for a large class of singular toric varieties.

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Section: Conic Sequence Of a Convex Polytopementioning

confidence: 79%

Conic decomposition of a toric variety and its application to cohomology

Park¹,

Song²

2021

Preprint

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“…We define the equivariant cohomology of simplicial GKM graph similarly to [GZ01, Section 1.7] and compare it with the equivariant cohomology of the corresponding simplicial GKM orbifold complexes. We prove the Thom isomorphism for equivariant cohomology and K-theory for general orbifold bundles (Theorem 2.16) which generalises the Thom isomorphism in [SS21,Section 2]. This helps to generalize the main result [HHH05, Theorem 2.3] to the category of G-spaces equipped with a stratification…”

Section: Introductionmentioning

confidence: 75%

“…The action of G on V i commutes with the action of G i for all i ∈ A. Then G induces an action on V i G i = U i ⊂ B and the composition map φ i • Pi : Ẽi → U i is also G invariant and this will induce the simple orbifold bundle P i : Ẽi G i → U i which is considered in [SS21]. Here we deal with general cases.…”

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

confidence: 99%

Integral generalized equivariant cohomologies of weighted Grassmann orbifolds

Koushik¹,

Sarkar²

2021

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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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“…So the natural question arises how to calculate the generalized G-equivariant cohomology theory E * G (Y ) of a GKM orbifold Y and more generally of a GKM orbifold simplicial complex. This question is answered partially in [SS21] for toric varieties associated to almost simple lattice polytopes. In this paper, we study the generalized G-equivariant cohomology theory…”

Section: Introductionmentioning

confidence: 99%

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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GKM theory for orbifold stratified spaces and application to singular toric varieties

Cited by 6 publications

References 38 publications

Conic decomposition of a toric variety and its application to cohomology

Conic decomposition of a toric variety and its application to cohomology

Integral generalized equivariant cohomologies of weighted Grassmann orbifolds

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

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