Integral generalized equivariant cohomologies of weighted Grassmann orbifolds

Abstract

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.

Torsion in the cohomology of blowups of quasitoric orbifolds

“…We prove that every WGr(k, n) is equivariantly homeomorphic to some Gr b (k, n) but not conversely. We derive that the definition of Gr b (k, n) generalizes the definition of weighted Grassmannian in [1] as well as weighted Grassmann orbifold in [5]. We show that Gr b (k, n) has an orbifold structure and call this space a weighted Grassmann orbifold.…”

“…, b m ), see Definition 2.5. We prove that Definition 2.5 broadens the class of weighted Grasmann orbifold discussed in [1] and [5]; see Proposition 2.8. We discuss the orbifold and q-CW complex structure of Gr b (k, n).…”

“…In [5], the author and Sarkar gave an affirmative answer to the first part of this question for divisive WGr(k, n). They computed the generalized T n -equivariant cohomology ring of divisive WGr(k, n), where T n is the standard compact abelian torus of dimension n. Moreover, they found an equivariant Schubert basis for the equivariant cohomology ring H * T n (Gr b (k, n); Z) as H * T n ({pt}; Z) module.…”

“…They studied the equivariant cohomology ring of WGr(k, n) with rational coefficients. In [5], the author and Sarkar gave another topological definition of WGr(k, n) and called them weighted Grassmann orbifolds. It has an orbifold structure and admits a q-CW complex structure similar to the Schubert cell decomposition of the Grassmann manifold.…”

“…They computed the generalized T n -equivariant cohomology ring of divisive WGr(k, n), where T n is the standard compact abelian torus of dimension n. Moreover, they found an equivariant Schubert basis for the equivariant cohomology ring H * T n (Gr b (k, n); Z) as H * T n ({pt}; Z) module. We consider the equivariant Schubert basis as a H * T n ({pt}; Z) module for the integral equivariant cohomology ring of divisive weighted Grassmann orbifold described in [5]. Then we explicitly calculate the equivariant structure constants with respect to that basis with integer coefficients.…”