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<p>In this paper, we prove the Thom isomorphism theorem for orbifold <i>G</i>-vector bundles for equivariant cohomology and equivariant <i>K</i>-theory with rational coefficients. We extend the main result of Harada et al. (Adv Math 197(1):198–221, 2005) to the category of <i>G</i>-spaces equipped with ‘singular invariant stratification.’ We introduce the concept of filtration of regular graphs and ‘simplicial graph complexes.’ Then, we introduce ‘simplicial GKM orbifold complexes’ and study some of their topological properties. We discuss the necessary conditions to confirm an invariant <i>q</i>-CW complex structure on a simplicial GKM orbifold complex. 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 compute the integral equivariant cohomology ring, equivariant <i>K</i>-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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年4月16日

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  • Koushik Brahma,
  • Soumen Sarkar

摘要

In this paper, 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 et al. (Adv Math 197(1):198–221, 2005) to the category of G-spaces equipped with ‘singular invariant stratification.’ We introduce the concept of filtration of regular graphs and ‘simplicial graph complexes.’ Then, we introduce ‘simplicial GKM orbifold complexes’ and study some of their topological properties. We discuss the necessary conditions to confirm an invariant q-CW complex structure on a simplicial GKM orbifold complex. 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 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.