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Reconstructing the orbit type stratification of a torus action from its equivariant cohomology

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Abstract

We investigate what information on the orbit type stratification of a torus action on a compact space is contained in its rational equivariant cohomology algebra. Regarding the (labelled) poset structure of the stratification, we show that equivariant cohomology encodes the subposet of ramified elements. For equivariantly formal actions, we also examine what cohomological information of the stratification is encoded. In the smooth setting, we show that under certain conditions—which in particular hold for a compact orientable manifold with discrete fixed point set—the equivariant cohomologies of the strata are encoded in the equivariant cohomology of the manifold.

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Funding

This work is part of a project funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 452427095. The second author is grateful to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support.

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  1. Philipps-Universität Marburg, Marburg, Germany

    Oliver Goertsches

  2. Max Planck Institute for Mathematics in Bonn, Bonn, Germany

    Leopold Zoller

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  1. Oliver Goertsches
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  2. Leopold Zoller
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Correspondence to Leopold Zoller.

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Goertsches, O., Zoller, L. Reconstructing the orbit type stratification of a torus action from its equivariant cohomology. J Algebr Comb 56, 799–822 (2022). https://doi.org/10.1007/s10801-022-01133-2

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  • DOI: https://doi.org/10.1007/s10801-022-01133-2

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