Equivariant Poincaré–Alexander–Lefschetz duality and the Cohen–Macaulay property

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.

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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

Volume 14, issue 3 (2014)

Christopher Allday, Matthias Franz and Volker Puppe

Abstract