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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Equivariant cohomology, syzygies and orbit structure


Authors: Christopher Allday, Matthias Franz and Volker Puppe
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 55N91; Secondary 13D02, 57P10
Published electronically: July 17, 2014
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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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Additional Information

Christopher Allday
Affiliation: Department of Mathematics, University of Hawaii, 2565 McCarthy Mall, Honolulu, Hawaii 96822
Email: chris@math.hawaii.edu

Matthias Franz
Affiliation: Department of Mathematics, University of Western Ontario, London, Ontario N6A5B7 Canada
Email: mfranz@uwo.ca

Volker Puppe
Affiliation: Fachbereich Mathematik, Universität Konstanz, 78457 Konstanz, Germany
Email: volker.puppe@uni-konstanz.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06165-5
PII: S 0002-9947(2014)06165-5
Received by editor(s): October 22, 2012
Received by editor(s) in revised form: March 8, 2013
Published electronically: July 17, 2014
Additional Notes: The second author was partially supported by an NSERC Discovery Grant.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.