Equivariant cohomology of odd-dimensional complex quadrics from a combinatorial point of view

TL;DR

This paper computes the torus equivariant cohomology ring of odd-dimensional complex quadrics using GKM graph combinatorics, revealing generators, relations, and connections to even-dimensional cases.

Contribution

It introduces a combinatorial approach to determine the ring structure of equivariant cohomology for odd-dimensional complex quadrics, highlighting new generators and relations.

Findings

Graph equivariant cohomology generated by three subgraph types

Identified four relations among generators

Established links between odd- and even-dimensional quadrics

Abstract

This paper aims to determine the ring structure of the torus equivariant cohomology of odd-dimensional complex quadrics by computing the graph equivariant cohomology of their corresponding GKM graphs. We show that its graph equivariant cohomology is generated by three types of subgraphs in the GKM graph, which are subject to four different types of relations. Furthermore, we consider the relationship between the two graph equivariant cohomology rings induced by odd- and even-dimensional complex quadrics.