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Braid groups and discrete diffeomorphisms of the punctured disk

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Abstract

We show that the group cohomology of the diffeomorphisms of the disk with n punctures has the cohomology of the braid group of n strands as the summand. As an application of this method, we also prove that there is no cohomological obstruction to lifting the “standard” embedding \(\mathrm {Br}_{2g+2}\hookrightarrow \mathrm {Mod}_{g,2}\) to a group homomorphism between diffeomorphism groups.

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Notes

  1. We borrowed this notation from [6].

  2. 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.

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Acknowledgements

I would like to thank Søren Galatius for his support and helpful discussions during this project. I would like to thank Kathryn Mann for introducing me to the realization problems and her interest in 1.6. I am indebted to Alexander Kupers for reading the first draft of this work. I also like to thank Jeremy Miller, Johannes Ebert, Ricardo Andrade and Jonathan Bowden for helpful discussions. Bena Tshishiku and Nick Salter provided me with an earlier draft of their paper for which I am very grateful. This project was partially supported by NSF grant DMS-1405001. Finally I would like thank the referee for his/her careful reading and many helpful suggestions.

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  1. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208, USA

    Sam Nariman

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Nariman, S. Braid groups and discrete diffeomorphisms of the punctured disk. Math. Z. 288, 1255–1271 (2018). https://doi.org/10.1007/s00209-017-1933-9

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