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Configuration Spaces pp 325–360Cite as

Syzygies in Equivariant Cohomology for Non-abelian Lie Groups

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Part of the book series: Springer INdAM Series ((SINDAMS,volume 14))

Abstract

We extend the work of Allday–Franz–Puppe on syzygies in equivariant cohomology from tori to arbitrary compact connected Lie groups G. In particular, we show that for a compact orientable G-manifold X the analogue of the Chang–Skjelbred sequence is exact if and only if the equivariant cohomology of X is reflexive, if and only if the equivariant Poincaré pairing for X is perfect. Along the way we establish that the equivariant cohomology modules arising from the orbit filtration of X are Cohen–Macaulay. We allow singular spaces and introduce a Cartan model for their equivariant cohomology. We also develop a criterion for the finiteness of the number of infinitesimal orbit types of a G-manifold.

The author was supported by an NSERC Discovery Grant.

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Notes

  1. 1.

    Using Propositions 5 and 7, one can show that H ∗ G​(X) is isomorphic to the equivariant Borel–Moore homology H BM, ∗ G(X) as introduced by Edidin and Graham [16, Sect. 2.8].

  2. 2.

    The local contractability assumptions made there were needed to ensure that singular and Alexander–Spanier cohomology coincide. This is not necessary in our setting, cf. [4, Remark 2.17].

  3. 3.

    The stronger assumption of finitely many orbit types made in [4, Sect. 2.1] only serves to allow the use of singular cohomology for the quotient X∕T instead of Alexander–Spanier cohomology.

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Acknowledgements

I would like to thank Chris Allday, Andrey Minchenko, Volker Puppe, Reyer Sjamaar and Andrzej Weber for helpful discussions, and the organizers of the conference “Configuration Spaces” for creating a stimulating environment in an extraordinary place. I am also indebted to the referee for a very careful reading and for several valuable suggestions and corrections.

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Authors and Affiliations

  1. Department of Mathematics, University of Western Ontario, London, ON, N6A 5B7, Canada

    Matthias Franz

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  1. Matthias Franz
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Correspondence to Matthias Franz .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Filippo Callegaro

  2. Department of Mathematics, University of Rochester, Rochester, USA

    Frederick Cohen

  3. Dipartimento di Matematica, Università di Roma La Sapienza, Roma, Italy

    Corrado De Concini

  4. Department of Mathematics, University of Bremen, Bremen, Germany

    Eva Maria Feichtner

  5. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Giovanni Gaiffi

  6. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Mario Salvetti

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Franz, M. (2016). Syzygies in Equivariant Cohomology for Non-abelian Lie Groups. In: Callegaro, F., Cohen, F., De Concini, C., Feichtner, E., Gaiffi, G., Salvetti, M. (eds) Configuration Spaces. Springer INdAM Series, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-31580-5_14

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