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Fox–Neuwirth cells, quantum shuffle algebras, and the homology of type-B Artin groups

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Abstract

In this paper, we will develop a family of braid representations of Artin groups of type B from braided vector spaces, and identify the homology of these groups with these coefficients with the cohomology of a specific bimodule over a quantum shuffle algebra. As an application, we give a complete characterization of the homology of type-B Artin groups with coefficients in one-dimensional braid representations over a field of characteristic 0. We will also discuss two different approaches to this computation: the first method extends a computation of the homology of braid groups due to Ellenberg–Tran–Westerland by means of induced representation, while the second method involves constructing a cellular stratification for configuration spaces of the punctured complex plane.

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Notes

  1. In this paper, we reserve the use of the notation \(B_n\) for the nth Artin group of type B. The nth braid group will be denoted by \(A_n\), due to its relation to the Artin groups of type A.

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Acknowledgements

The author would like to acknowledge Craig Westerland for initiating the subject of study of this paper and helpful discussions. We also appreciate Calista Bernard for useful suggestions, and Alexander Voronov for providing insights about the Hochschild chain complex and Hochschild homology. Finally, we thank anonymous referees for helpful comments.

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  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Anh Trong Nam Hoang

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Hoang, A.T.N. Fox–Neuwirth cells, quantum shuffle algebras, and the homology of type-B Artin groups. Math. Z. 306, 57 (2024). https://doi.org/10.1007/s00209-024-03445-4

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