Equivariant cohomology, syzygies and orbit structure

Allday, Christopher; Franz, Matthias; Puppe, Volker

doi:10.1090/S0002-9947-2014-06165-5

Equivariant cohomology, syzygies and orbit structure
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by Christopher Allday, Matthias Franz and Volker Puppe PDF

Trans. Amer. Math. Soc. 366 (2014), 6567-6589 Request permission

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 $\mathrm {Ext}$ module involving a suitably defined equivariant homology of $X$.

One consequence is that the GKM method for computing equivariant cohomology applies to a Poincaré duality space if and only if the equivariant Poincaré pairing is perfect.

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