Topological Hochschild homology of Thom spectra and the free loop space
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 \[ THH(MG) \simeq MG \wedge BBG_+ , \] where \(G\) denotes a stabilized Lie group \(O\), \(SO\), \(Spin\), \(U\), or \(Sp\). The theorem is also applied to the Eilenberg-MacLane spectra \(H{\mathbf{Z}}/p\) and \(H{\mathbf{Z}}\), recovering a result of \textit{M. Bökstedt} [``The topological Hochschild homology of \({\mathbf{Z}}\) and \({\mathbf{Z}}/p\), preprint (1990)]. Thus, there are stable equivalences \[ \begin{aligned} & THH(H{\mathbf{Z}}/p) \simeq H{\mathbf{Z}}/p \wedge \Omega(S^3)_+ \\ & THH(H{\mathbf{Z}}) \simeq H{\mathbf{Z}} \wedge \Omega(S^3 \langle 3 \rangle)_+, \end{aligned} \] where \(S^3 \langle 3 \rangle\) denotes the 3-connected cover of \(S^3\). The authors work through symmetric monoidal models of the category of spaces over \(BF\) as well as strong symmetric monoidal Thom spectrum functors.
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