The syzygy order of big polygon spaces

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 $H^{\ast }(BT)$. 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.