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Thom spectra, higher $\mathrm{THH}$ and tensors in $\infty$–categories

Nima Rasekh, Bruno Stonek and Gabriel Valenzuela

Algebraic & Geometric Topology 22 (2022) 1841–1903

Let f : G Pic (R) be a map of E–groups, where Pic (R) denotes the Picard space of an E–ring spectrum R. We determine the tensor X RMf of the Thom ER–algebra Mf with a space X; when X is the circle, the tensor with X is topological Hochschild homology over R. We use the theory of localizations of –categories as a technical tool: we contribute to this theory an –categorical analogue of Day’s reflection theorem about closed symmetric monoidal structures on localizations, and we prove that, for a smashing localization L of the –category of presentable –categories, the free L–local presentable –category on a small simplicial set K is given by presheaves on K valued on the L–localization of the –category of spaces.

If X is a pointed space, a map g: A B of E–ring spectra satisfies X–base change if X B is the pushout of A X A along g. Building on a result of Mathew, we prove that if g is étale then it satisfies X–base change provided X is connected. We also prove that g satisfies X–base change provided the multiplication map of B is an equivalence. Finally, we prove that, under some hypotheses, the Thom isomorphism of Mahowald cannot be an instance of S0–base change.

Thom spectra, topological Hochschild homology, $E_\infty$ ring spectra, Thom isomorphism, topological K–theory
Mathematical Subject Classification
Primary: 55P43
Secondary: 18D20, 55N20
Received: 8 April 2020
Revised: 29 December 2020
Accepted: 29 March 2021
Published: 10 October 2022
Nima Rasekh
Max Planck Institute for Mathematics
Bruno Stonek
University of Warsaw
Gabriel Valenzuela
Max Planck Institute for Mathematics