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Topological coHochschild homology and the homology of free loop spaces


We study the homology of free loop spaces via techniques arising from the theory of topological coHochschild homology (coTHH). Topological coHochschild homology is a topological analogue of the classical theory of coHochschild homology for coalgebras. We produce new spectrum-level structure on coTHH of suspension spectra as well as new algebraic structure in the coBökstedt spectral sequence for computing coTHH. These new techniques allow us to compute the homology of free loop spaces in several new cases, extending known calculations.

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The authors express their gratitude to the organizers of the Women in Topology II Workshop and the Banff International Research Station, where this collaboration began. We thank Vigleik Angeltveit, Ben Antieau, David Chan, Paul Goerss, Kathryn Hess, Sarah Klanderman, Maximilien Péroux, Emily Riehl, and Stephanie Ziegenhagen for helpful conversations related to this work. The authors also thank an anonymous referee for helpful comments. This research was supported by the National Science Foundation [DMS-1710534 and DMS-2104300 to Bohmann, DMS-1810575 to Gerhardt, and DMS-1811278 to Shipley]. Some of this work was done while the second author was in residence at the Mathematical Sciences Research Institute in Berkeley, CA (supported by the National Science Foundation under Grant DMS-1440140) during the Spring 2020 semester. The first and third authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the program ‘Homotopy harnessing higher structures’ during which some of the work on this paper was carried out. Work during this program was supported by EPSRC Grant no EP/K032208/1.

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Correspondence to Anna Marie Bohmann.

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Bohmann, A.M., Gerhardt, T. & Shipley, B. Topological coHochschild homology and the homology of free loop spaces. Math. Z. 301, 411–454 (2022).

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  • Topological Hochschild homology
  • Coalgebra
  • Free loop spaces

Mathematics Subject Classification

  • Primary: 55P35
  • 16T15
  • Secondary: 55P43
  • 13D03
  • 16T05
  • 55T99