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

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.

💡 Research Summary

The paper determines the integral torus‑equivariant cohomology rings of odd‑dimensional complex quadrics (Q_{2n-1}) by exploiting GKM (Goresky‑Kottwitz‑MacPherson) theory. The authors first observe that the standard maximal torus (T^n) acts effectively on (Q_{2n-1}) and that the fixed‑point set consists of the (2n) coordinate points (