Published by De Gruyter June 13, 2024

Homology of configuration spaces of surfaces modulo an odd prime

Andrea Bianchi and Andreas Stavrou

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2024-0029

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Abstract

For a compact orientable surface Σ g , 1 of genus 𝑔 with one boundary component and for an odd prime number 𝑝, we study the homology of the unordered configuration spaces C ∙ ( Σ g , 1 ) : = ∐ n ≥ 0 C n ( Σ g , 1 ) with coefficients in F p . We describe H ∗ ⁢ ( C ∙ ⁢ ( Σ g , 1 ) ; F p ) as a bigraded module over the Pontryagin ring H ∗ ⁢ ( C ∙ ⁢ ( D ) ; F p ) , where 𝐷 is a disc, and compute in particular the bigraded dimension over F p . We also consider the action of the mapping class group Γ g , 1 and prove that the mod-𝑝 Johnson kernel K g , 1 ⁢ ( p ) ⊆ Γ g , 1 is the kernel of the action on H ∗ ⁢ ( C ∙ ⁢ ( Σ g , 1 ; F p ) ) .

Funding source: Horizon 2020 Framework Programme

Award Identifier / Grant number: 772960

Funding source: Danmarks Grundforskningsfond

Award Identifier / Grant number: DNRF151

Funding source: Engineering and Physical Sciences Research Council

Award Identifier / Grant number: 2261124

Funding statement: A. Bianchi was supported by the European Research Council under the European Union’s Horizon2020 research and innovation programme (grant agreement No. 772960), and by the Danish National Research Foundation through the Copenhagen Centre for Geometry and Topology (DNRF151). A. Stavrou was funded by a studentship of the Engineering and Physical Sciences Research Council (project reference 2261124).

Acknowledgements

The first author thanks Lorenzo Guerra for a discussion about divided power algebras, and Jeremy Miller for a discussions about homology operations on configuration spaces. Both authors thank Oscar Randal-Williams for several discussions on the topic and for his supervision of the second author. Both authors thank Zachary Himes, Jeremy Miller, Jan Steinebrunner, Nathalie Wahl, Adela Zhang and the referee for comments on a first draft of the article. The article was prepared while A. Bianchi was affiliated with the University of Copenhagen and A. Stavrou with the University of Cambridge.

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Received: 2023-08-03

Revised: 2024-04-22

Published Online: 2024-06-13

Published in Print: 2024-08-01

Homology of configuration spaces of surfaces modulo an odd prime

Abstract

Acknowledgements

References

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