Volume 14, issue 3 (2014)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Equivariant Poincaré–Alexander–Lefschetz duality and the Cohen–Macaulay property

Christopher Allday, Matthias Franz and Volker Puppe

Algebraic & Geometric Topology 14 (2014) 1339–1375
Abstract

We prove a Poincaré–Alexander–Lefschetz duality theorem for rational torus-equivariant cohomology and rational homology manifolds. We allow non-compact and non-orientable spaces. We use this to deduce certain short exact sequences in equivariant cohomology, originally due to Duflot in the differentiable case, from similar, but more general short exact sequences in equivariant homology. A crucial role is played by the Cohen–Macaulayness of relative equivariant cohomology modules arising from the orbit filtration.

Keywords
torus actions, homology manifolds, equivariant homology, equivariant cohomology, Atiyah–Bredon complex, Poincaré–Alexander–Lefschetz duality, Cohen–Macaulay modules
Mathematical Subject Classification 2010
Primary: 55N91
Secondary: 13C14, 57R91
References
Publication
Received: 30 March 2013
Revised: 16 September 2013
Accepted: 30 September 2013
Published: 7 April 2014
Authors
Christopher Allday
Department of Mathematics
University of Hawaii
2565 McCarthy Mall
Honolulu, HI 96822
USA
Matthias Franz
Department of Mathematics
University of Western Ontario
London, ON N6A 5B7
Canada
http://www.math.uwo.ca/~mfranz/
Volker Puppe
Fachbereich Mathematik und Statistik
Universität Konstanz
D-78457 Konstanz
Germany