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

Abstract

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 .