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Dynamical and cohomological obstructions to extending group actions

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Abstract

Motivated by a question of Ghys, we study obstructions to extending group actions on the boundary \(\partial M\) of a 3-manifold to a \(C^0\)-action on M. Among other results, we show that for a 3-manifold M, the \(S^1 \times S^1\) action on the boundary does not extend to a \(C^0\)-action of \(S^1 \times S^1\) via homeomorphisms that are isotopic to the identity as a discrete group on M, except in the trivial case \(M \cong D^2 \times S^1\).

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Notes

  1. For a topological group G acting on a topological space X, the homotopy quotient is denoted by \(X/\!\!/G\) and is given by \(X\times _G \mathrm {E}G\) where \(\mathrm {E}G\) is a contractible space on which G acts freely and properly discontinuously.

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Acknowledgements

SN was partially supported by NSF Grant DMS-1810644 and he would like to thank the Isaac Newton Institute for Mathematical sciences for support and hospitality during the program “homotopy harnessing higher structures”. He thanks Oscar Randal-Williams and Søren Galatius for helpful discussions. KM was partially supported by NSF Grant DMS-1606254. We also thank Sander Kupers for his comments on the early draft of this paper, and the referee for many helpful remarks.

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Authors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, NY, 14850, USA

    Kathryn Mann

  2. Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    Sam Nariman

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  1. Kathryn Mann
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Correspondence to Sam Nariman.

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Communicated by Thomas Schick.

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Mann, K., Nariman, S. Dynamical and cohomological obstructions to extending group actions. Math. Ann. 377, 1313–1338 (2020). https://doi.org/10.1007/s00208-020-01989-4

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