Topological Hochschild homology of Thom spectra and the free loop space

By Andrew Justin Blumberg, R. Cohen, Christian Schlichtkrull. 2008

71 citations recorded by the source corpus.

We describe the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps f : X! BF, where BF denotes the classifying space for stable spherical fibrations. To do this, we consider sym- metric monoidal models of the category of spaces over BF and corresponding strong symmetric monoidal Thom spectrum functors. Our main result identi- fies the topological Hochschild homology as the Thom spectrum of a certain stable bundle over the free loop space L(BX). This leads to explicit calcula- tions of the topological Hochschild homology for a large class of ring spectra, including all of the classical cobordism spectra MO, MSO, MU, etc., and the Eilenberg-Mac Lane spectra HZ/p and HZ.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/35966b425b2f437fb0ac98cba9cf27af6fc52df6

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Citations in the corpus

Global Homotopy Theory (2018, background): “The category of L-spaces has been much studied, for example in [17, 19, 91, 98]; in some of these source the symbol L refers to the linear isometries operad, so the monoid we denote L is then the monoid L(1) of unary operations.”
Diagram spaces and symmetric spectra (2011, background): “This is the flatness criterion used in [3, 31], but it is not the right notion in the present setting of cofibrantly generated model categories.”
An ∞‐categorical approach to R‐line bundles, R‐module Thom spectra, and twisted R‐homology (2014, unknown): “In order to make this outline precise, the companion paper used the technology of ∗ -modules [ 3 , 4 ], which are a symmetric monoidal model for the category of spaces such that monoids are precisely A ∞ -spaces and commutative monoids are precisely E ∞ -spaces.”
Units of ring spectra and Thom spectra (2008, background): “ the second author [Blum05]. In this paper we present a streamlined exposition covering the part of the theory we need for our applications. The interested reader should consult the forthcoming paper [BCS08] for further discussion and in particular proofs of the foundational theorems stated below. 5”
The Local Structure of Algebraic K-Theory (2012, background): “These equivalences arise as chains of natural equivalences respecting the E∞-structure (see references to preprints in [20]).”
Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory (2014, unknown): “A detailed presentation of the theory (along with complete proofs) has appeared in [7] (and see also [19]).”