Torsion in the cohomology of blowups of quasitoric orbifolds

Koushik, Brahma,; Sarkar, Soumen; Sau, Subhankar

doi:10.1016/j.topol.2021.107666

Cited by 4 publications

(5 citation statements)

References 21 publications

(42 reference statements)

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“…Thus the simplicial graph complex associated to K is the same as the simplicial GKM graph complex G in Example 2.27. Now if c i divides c i−1 for all i ∈ {1, 2, 3} then by [BS,Theorem 3.19], one can show that K has a T 4 -invariant cell-structure and H odd (K; Z) = 0. Therefore, by Theorem 4.6…”

Section: Generalized Cohomologies Of Simplicial Gkm Orbifold Complexesmentioning

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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Section: Generalized Cohomologies Of Simplicial Gkm Orbifold Complexesmentioning

confidence: 99%

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

Koushik¹,

Sarkar²

2023

Preprint

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“…The idea of blowup of a polytope along a face is studied in [GP13] and [BSS21]. The vertex (zero dimensional face) cut of a polytope is a special case of blowup of polytope.…”

Section: (Quasi)toric Manifolds Over a Vertex Cut Of A Finite Product...mentioning

confidence: 99%

“…By edge cut we mean a blowup along an edge(dimension one face), see [BSS21] for the details of blowup of polytope along a face. In Section 6, we study the (quasi)toric manifold over an edge cut of a cube where the edge consists of the vertices tṽ 1 , ṽ2 u and the characteristic function is defined as in (6.1) satisfying (1.2).…”

Section: Introductionmentioning

confidence: 99%

On cohomology of quasitoric manifolds over a vertex cut of a finite product of simplices

Sarkar¹,

Sau²

2021

Preprint

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In this paper, we give the necessary and sufficient condition that when a quasitoric manifold over a product of two simplices is a toric manifold. Moreover, we show that a toric manifold over a finite product of simplices satisfies this necessary condition. Then we study some combinatorial properties of (quasi)toric manifolds over a vertex cut of a finite product of simplices. We discuss their integral cohomology rings with possibly minimal generators and show several relations among the product of these generators. In the course of generalization, we study some of the combinatorial properties of (quasi)toric manifolds over an edge cut of an n-cube. We also discuss the integral cohomology rings of (quasi)toric manifolds over an edge cut of a 3-dimensional cube. Contents1. Introduction 1 2. Cohomology of (quasi)toric manifold over a finite product of simplices 3 3. Toric manifold over a finite product of simplices 8 4. (Quasi)toric manifolds over a vertex cut of a finite product of simplices 18 5. Cohomology rings of (quasi)toric manifolds over a vertex cut of a finite product of simplices 30 6. (Quasi)toric manifolds over an edge cut of an n-cube 39 References 44

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“…In algebraic topology, [GP13] used blowups for resolution of singularities of four dimensional quasitoric orbifolds. In this article, we use the techniques of blowup of quasitoric orbifold discussed in [BSS21] for resolution of singularities of a quasitoric orbifold.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, we recall the notion of quasitoric manifold and quasitoric orbifold following [DJ91], [BP02] and [PS10]. Then we recall the notion of blowup of a simple polytope as well as blowup of a quasitoric orbifold following [BSS21]. We discuss the orbifold singularity on any face in the orbit space of a quasitoric orbifold.…”

Section: Introductionmentioning

confidence: 99%

Resolution of singularities of quasitoric orbifolds and equivariant cobordism of quasi-contact toric manifolds

Koushik¹,

Sarkar²,

Sau³

2022

Preprint

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Torsion in the cohomology of blowups of quasitoric orbifolds

Cited by 4 publications

References 21 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

On cohomology of quasitoric manifolds over a vertex cut of a finite product of simplices

Resolution of singularities of quasitoric orbifolds and equivariant cobordism of quasi-contact toric manifolds

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