Equivariant cohomology, syzygies and orbit structure

Abstract

Let X be a "nice" space with an action of a torus T. We consider the Atiyah-Bredon sequence of equivariant cohomology modules arising from the filtration of X by orbit dimension. We show that a front piece of this sequence is exact if and only if the H^*(BT)-module H_T^*(X) is a certain syzygy. Moreover, we express the cohomology of that sequence as an Ext module involving a suitably defined equivariant homology of X. One consequence is that the GKM method for computing equivariant cohomology applies to a Poincare duality space if and only if the equivariant Poincare pairing is perfect.Comment: 23 pages. The former Section 6 has been substantially expanded and is now a separate paper, see arXiv:1303.1146. Several minor change

Equivariant cohomology of ${(\mathbb Z_{2})^{r}}$-manifolds and syzygies

“…In this note we consider actions of a 2-torus G = (Z 2 ) r and coefficients in the field F 2 of characteristic 2. All major analogous results of [2] and [3], in particular Theorem 1.1 above, turn out to be true in this setting. Nevertheless, some of them require new methods of proof, basically because, in contrast to T = (S 1 ) r , G = (Z 2 ) r has only finitely many subgroups and because the field F 2 has only finitely many elements.…”

“…This class contains the real versions of the 'big polygon spaces' defined and considered by M.Franz in [12]. We calculate the equivariant cohomology with F 2 -coefficients, which in many examples turns out to be torsion-free but not free and realizes all orders of syzygies, which are in concordance with the restrictions proved in [4]. The final results for the real versions are analogous to those for the big polynomial spaces in [12], where (S 1 ) r -actions and rational coefficients are considered, but we consider also a wider class of manifolds here and the point of view as well as the method of proof, for which it is essential to consider equivariant cohomology for diversbut related -groups, are quite different.2010 Mathematics Subject Classification.…”

“…Nevertheless, some of them require new methods of proof, basically because, in contrast to T = (S 1 ) r , G = (Z 2 ) r has only finitely many subgroups and because the field F 2 has only finitely many elements. This is carried out in a so far unpublished manuscript [4], even for the not quite analogous case of G = (Z p ) r -actions and coefficients in a field k of characteristic p > 0, p an odd prime, which is somewhat more involved than the p = 2 case.…”

Symmetric Products of Equivariantly Formal Spaces

“…In a different direction, the notion of equivariant formality has been extended to that of a syzygy in equivariant cohomology by Allday-Franz-Puppe [2] (G a torus) and Franz [10] (G a compact connected Lie group). Let r be the rank of such a G, so that H * (BG; R) is a polynomial algebra in r variables of even degrees.…”

Polynomial assignments

“…By Proposition 3.3, equivariant formality implies that H T (M) is a free S(t * )-module. As pointed out in [1], the Chang-Skjelbred Theorem is valid when the equivariant formality condition is replaced by the milder condition that H T (M) be a reflexive S(t * )-module.…”