# The syzygy order of big polygon spaces

## TL;DR

This paper determines the precise syzygy order of the equivariant cohomology of big polygon spaces, revealing new algebraic properties related to the length vector defining these manifolds.

## Contribution

It provides an exact calculation of the syzygy order for big polygon spaces' equivariant cohomology based on their length vectors, using a novel characterization of syzygies.

## Key findings

- Exact syzygy order expressed in terms of length vector
- Refined characterization of syzygies in terms of linearly independent elements in H^2(BT)
- Application to understanding torsion-free and reflexive properties

## Abstract

Big polygon spaces are compact orientable manifolds with a torus action whose equivariant cohomology can be torsion-free or reflexive without being free as a module over $H^*(BT)$. We determine the exact syzygy order of the equivariant cohomology of a big polygon space in terms of the length vector defining it. The proof uses a refined characterization of syzygies in terms of certain linearly independent elements in $H^2(BT)$ adapted to the isotropy groups occurring in a given $T$-space.