On Integral Cohomology of Certain Orbifolds

Bahri, Abbas; Notbohm, Dietrich; Sarkar, Soumen; Song, Jongbaek

doi:10.1093/imrn/rny283

Cited by 15 publications

(8 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The main tool of the current paper, namely the Mayer-Vietoris sequence, is fundamentally different from, and simpler than, the techniques developed in [13]. The other notable difference is that the present paper is concerned solely with equivariant K-theory, whereas [13] is one of a number of papers [3,4,5,10] to consider other complex oriented equivariant cohomology theories.…”

Section: Background and Notationmentioning

confidence: 96%

Mayer–Vietoris sequences and equivariant $K$-theory rings of toric varieties

Holm

Williams

2019

Homology, Homotopy and Applications

View full text Add to dashboard Cite

We apply a Mayer-Vietoris sequence argument to identify the Atiyah-Segal equivariant complex K-theory rings of certain toric varieties with rings of integral piecewise Laurent polynomials on the associated fans. We provide necessary and sufficient conditions for this identification to hold for toric varieties of complex dimension 2, including smooth and singular cases. We prove that it always holds for smooth toric varieties, regardless of whether or not the fan is polytopal or complete. Finally, we introduce the notion of fans with "distant singular cones," and prove that the identification holds for them. The identification has already been made by Hararda, Holm, Ray and Williams in the case of divisive weighted projective spaces; in addition to enlarging the class of toric varieties for which the identification holds, this work provides an example in which the identification fails. We make every effort to ensure that our work is rich in examples.

show abstract

Section: Background and Notationmentioning

confidence: 96%

Mayer–Vietoris sequences and equivariant $K$-theory rings of toric varieties

Holm

Williams

2019

Homology, Homotopy and Applications

View full text Add to dashboard Cite

show abstract

“…The aim of this section is to study the Poincaré polynomial of certain toric varieties. The main idea is to use a retraction sequence of a polytope introduced in [2,3,16]. We first begin by reviewing some facts on projective toric varieties associated with lattice polytopes and their properties which are necessary to our aim.…”

Section: Backgrounds: Polytopes and Projective Toric Varietiesmentioning

confidence: 99%

“…Indeed, there exist two combinatorially equivalent lattice polytopes for which the associated toric varieties have different Betti numbers, see [15]. There are several approaches to computing the Betti numbers of singular toric varieties, e.g., using spectral sequences (see [9,12] and [8, §12.3]), Bia lynicki-Birula decomposition [4], retraction sequences on polytopes [2,3,16].…”

Section: Introductionmentioning

confidence: 99%

Generic torus orbit closures in Schubert varieties

Lee¹,

Masuda²

2020

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

The closure of a generic torus orbit in the flag variety G/B of type A is known to be a permutohedral variety and its Poincaré polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincaré polynomial of a generic torus orbit closure in a Schubert variety in G/B. When the generic torus orbit closure in a Schubert variety is smooth, its Poincaré polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.

show abstract

“…0] ∈ CP n χ , which is same as |G ∆ n (v i )| defined from the R-characteristic pair (∆ n , λ), where v i = F 1 ∩ · · · ∩ F i1 ∩ F i+1 ∩ · · · F n+1 . Now, following the proof of [BNSS,Proposition 4.5], one can always find retraction sequences {(B j , E j , b j )} n+1 j=1 which satisfies the assumption of Theorem 3.7. Hence, we conclude that the (co)homology of any weighted projective space is torsion free and concentrated in even degrees.…”

Section: Building Sequences and Homology Of Toric Orbifoldsmentioning

confidence: 99%

Infinite families of equivariantly formal toric orbifolds

2018

Self Cite

View full text Add to dashboard Cite

The simplicial wedge construction on simplicial complexes and simple polytopes has been used by a variety of authors to study toric and related spaces, including non-singular toric varieties, toric manifolds, intersections of quadrics and more generally, polyhedral products. In this paper we extend the analysis to include toric orbifolds. Our main results yield infinite families of toric orbifolds, derived from a given one, whose integral cohomology is free of torsion and is concentrated in even degrees, a property which might be termed integrally equivariantly formal. In all cases, it is possible to give a description of the cohomology ring and to relate it to the cohomology of the original orbifold. Though Davis and Januszkiewicz [DJ91] introduced the name toric manifolds, in recent literature they are sometimes called quasitoric manifolds.

show abstract

On Integral Cohomology of Certain Orbifolds

Cited by 15 publications

References 24 publications

Mayer–Vietoris sequences and equivariant $K$-theory rings of toric varieties

Mayer–Vietoris sequences and equivariant $K$-theory rings of toric varieties

Generic torus orbit closures in Schubert varieties

Infinite families of equivariantly formal toric orbifolds

Contact Info

Product

Resources

About