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

TL;DR

This paper introduces a new class of simplicial GKM orbifold complexes, explores their topological properties, and computes their integral equivariant cohomology, K-theory, and cobordism rings, extending existing theorems to singular stratified spaces.

Contribution

It defines simplicial GKM orbifold complexes and provides a combinatorial framework for their equivariant cohomology, K-theory, and cobordism rings, extending classical results to singular orbifold settings.

Findings

Simplicial GKM orbifold complexes are equivariantly formal under certain conditions.

Explicit computation of equivariant cohomology, K-theory, and cobordism rings for divisive complexes.

Extension of Thom isomorphism and Harada-Henriques-Holm results to orbifold G-spaces with singular stratification.

Abstract

In this paper, we define `simplicial GKM orbifold complexes' and study some of their topological properties. 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 $q$-CW complex 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, and a divisive simplicial GKM orbifold complex is integrally equivariantly formal. 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 a basis of the integral generalized equivariant cohomology of a divisive simplicial GKM orbifold complex.