Integral generalized equivariant cohomologies of weighted Grassmann orbifolds

TL;DR

This paper defines weighted Grassmann orbifolds, explores their topological invariants, and computes their equivariant cohomology, K-theory, and cobordism rings, providing new tools for understanding their geometric and algebraic structures.

Contribution

It introduces weighted Grassmann orbifolds, analyzes their cell structures and singularities, and computes their equivariant cohomology, K-theory, and cobordism rings with various coefficients.

Findings

Weighted Grassmann orbifolds have invariant q-cell structures.

The integral cohomology of these orbifolds can be torsion-free under certain conditions.

Explicit formulas for structure constants in equivariant cohomology are provided.

Abstract

We introduce a new definition of weighted Grassmann orbifolds. We study their several invariant $q$-cell structures and the orbifold singularities on these $q$-cells. We discuss when the integral cohomology of a weighted Grassmann orbifold has no $p$-torsion. We compute the equivariant $K$-theory ring of weighted Grassmann orbifolds with rational coefficients. We introduce divisive weighted Grassmann orbifolds and show that they have invariant cell structures. We calculate the equivariant cohomology ring, equivariant $K$-theory ring and equivariant cobordism ring of a divisive weighted Grassmann orbifold with integer coefficients. We discuss how to compute the weighted structure constants for the integral equivariant cohomology ring of a divisive weighted Grassmann orbifold.