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 * (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.

Syzygies in equivariant cohomology in positive characteristic

“…4-6] can actually be carried out over any field. Hence these examples remain valid for T -equivariant cohomology with coefficients in k. Moreover, the argument in [15,Sec. 3] is purely algebraic and again works over any field (see also Proposition 8.4).…”

“…5.12] of Proposition 8.13. The exact syzygy order has been determined in [15,Thm. 3.2] for all big polygon spaces.…”

“…For p = 2 another family of examples is given by the real versions of big polygon spaces studied in [22] and [15,Sec. 5], with analogous properties.…”

“…It is clear that (2) implies (2) * since the R G -free modules F i appearing in the exact sequence (2.13) Analogously to Proposition 8.4, one can relax condition (3) by using only sequences contained in a j-localizing subset S ⊂ H 1 (BG; F 2 ), see [15,Lemma 5.2].…”