Equivariant cohomology of (Z_2)^r - manifolds and syzygies

TL;DR

This paper computes the equivariant cohomology of certain (Z_2)^r-manifolds, including polygon and chain spaces, revealing torsion-free but non-free modules over the group cohomology, with implications for syzygy theory.

Contribution

It provides explicit calculations of equivariant cohomology for (Z_2)^r-manifolds and explores their module structure related to syzygies, extending understanding of these spaces.

Findings

Equivariant cohomology modules are often torsion free.

Modules are not necessarily free over the group cohomology.

Results relate to the notion of syzygy in algebraic topology.

Abstract

We consider closed manifolds, which occur as intersections of products of spheres of the same dimension with certain hyperplanes. Among those are the so called (big) polygon- and chain spaces. The equivariant cohomology with respect to natural actions of 2-tori is calculated and related to the notion of syzygy. It turns out that the equivariant cohomology module of these manifolds is often torsion free, but not free over the cohomology of the group. Coefficients are taken in the field with two elements.