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Homotopy invariants for via Koszul duality

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Abstract

We show that the integer cohomology rings of the moduli spaces of stable rational marked curves are Koszul. This answers an open question of Manin. Using the machinery of Koszul spaces developed by Berglund, we compute the rational homotopy Lie algebras of those spaces, and obtain some estimates for Betti numbers of their free loop spaces in case of torsion coefficients. We also prove and conjecture some generalisations of our main result.

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Acknowledgements

I am grateful to Yuri Ivanovich Manin for encouragement. I also wish to thank Greg Arone, Alexander Berglund, Alex Fink, Vincent Gélinas, Anton Khoroshkin, Natalia Iyudu, Vic Reiner, Pedro Tamaroff, and Bruno Vallette for useful discussions. The paper benefited from extraordinarily thorough peer review process, and I wish to offer my deep gratitude to the anonymous referee whose queries made me spell out the proof in great detail, leaving no stone unturned. Special thanks are due to Neil Strickland for providing a copy of [46], and to Geoffroy Horel for a discussion of results of [9].

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  1. Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René-Descartes, 67000, Strasbourg, France

    Vladimir Dotsenko

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Dotsenko, V. Homotopy invariants for via Koszul duality. Invent. math. 228, 77–106 (2022). https://doi.org/10.1007/s00222-021-01081-x

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