Topological Hochschild homology of Thom spectra and the free loop space
type: Item
instance of: scholarly article
title: Topological Hochschild homology of Thom spectra and the free loop space
zbMATH Open document ID: 1219.19006
author: Andrew J. Blumberg
author: Ralph L. Cohen
author: Christian Schlichtkrull
published in: Geometry \& Topology
publication date: 2010-07-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/0811.0553
review text: The paper under review identifies the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps \( f: X \to BF\). Here \(BF\) is the colimit of the classifying spaces \(BF(n)\), where \(F(n)\) is the topological monoid of base-point preserving homotopy equivalences of \(S^n\), the \(n\)-sphere. Letting \(EF(n)\) denote the one-sided bar construction \(B(*, \, F(n), \, S^n)\), the Thom space of a map \(f: X \to BF(n)\) is defined as the quotient space \[ T(f) = f^* EF(n)/X, \] where \(F^*EF(n)\) is the pullback of \(EF(n)\). For more details about the Lewis-May Thom spectrum functor, see \textit{L. G. Lewis jun., J. P. May} and \textit{M. Steinberger} [Equivariant stable homotopy theory. With contributions by J. E. McClure, Lect. Notes Math. 1213 (1986; Zbl 0611.55001)]. To describe the main theorem of the paper, let \(B\) be a connected space, \(L(B)\) its free loop space, and \(\Omega (B)\) the based loop space. When \(B\) is a homotopy associative and commutative \(H\)-space, \(L(B)\) is as well, and there is a (non-equivariant) equivalence of \(H\)-spaces \[ L(B) \overset{\simeq}{\longrightarrow} \Omega(B) \times B. \] Thus, \[ L(B^2F) \simeq \Omega (B^2F) \times B^2F \simeq BF \times B^2F. \] The Hopf fibration \(S^3 \to S^2\) induces a map \[ \eta : B^2F \simeq {\text{Map}}_*(S^2, \;B^4F) \to {\text{Map}}_*(S^3, \;B^4F) \simeq BF. \] Let \(f: X \to BF\) be the loop map associated to a map of based spaces \(Bf : BX \to B^2f\). Then there is a natural stable equivalence \[ THH (T(f)) \simeq T(L^{\eta}(Bf)), \] where \(L^{\eta}(Bf)\) denotes the composite \[ L(BX) \overset{L(Bf)}{\longrightarrow} L(B^2F) \simeq BF \times B^2F \overset{{\mathbf{1}}\times \eta}{\longrightarrow} BF \times {BF} BF. \] Moreover, if \(f\) is a 2-fold loop map, then there is a stable equivalence \[ THH(T(f)) \simeq T(f) \wedge T(\eta \circ Bf). \] These results are applied to the classical cobordism spectra \(MG\), yielding \[ THH(MG) \simeq MG \wedge BBG_+…
reviewed by: Jerry M. Lodder
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 55N20
Mathematics Subject Classification ID: 55P43
zbMATH DE Number: 5730921
zbMATH Keywords: topological Hochschild homology
zbMATH Keywords: Thom spectra
zbMATH Keywords: ring spectra
zbMATH Keywords: free loop space
MaRDI profile type: Publication
OpenAlex ID: W2125813633
arXiv ID: 0811.0553
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DOI: 10.2140/GT.2010.14.1165
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