The syzygy order of big polygon spaces

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 $H^{\ast }\left(BT\right)$. We determine the exact syzygy order of the equivariant cohomology of a big polygon space as a function of the length vector defining it. The proof uses a refined characterization of syzygies in terms of certain linearly independent elements in $H^2\left(BT\right)$ adapted to the isotropy groups occurring in a given $T$–space.