Generalized Thom spectra and their topological Hochschild homology. (English) Zbl 1430.19006

Basu, Samik; Sagave, Steffen; Schlichtkrull, Christian

J. Inst. Math. Jussieu 19, No. 1, 21-64 (2020).

Let \(\mathbf{R}\) be a structure ring spectrum, this has an underlying grouplike monoid of units \(\mathrm{GL}_{1}(\mathbf{R})\). Let \(X\) be a space, this spectrum represents the functor that assigns to \(X\) the units in the \(\mathbf{R}\)-cohomology classes \(\mathbf{R}^{0}(X)\). Next, given a map \(X\to \mathrm{BGL}{1}(\mathbf{R})\), this has associated an \(\mathbf{R}\)-module Thom spectrum \(M(f)\). The authors develop an analogous machinery in the setup of diagram spectra. An important feature is the multiplicative properties of their construction. As applications, they show several examples where known spectra arise a Thom spectrum of some map, for example: Theorem. Let \(\mathbf{R}\) be a symmetric ring spectrum where all of its homotopy groups are concentrated in even degrees. Let \(u_{1},\ldots , u_{n}\) be a sequence of homotopy classes with \(u_{i}\in \pi_{2i} (\mathbf{R})\). Then, there is a loop map \(f_{(u_{1},\ldots , u_{n})}:SU(n+1)\to BG_{hI}\) such that the homotopy type of the \(\mathbf{R}\)-module underlying the associated \(\mathbf{R}\)-algebra Thom spectrum \(T(f_{(u_{1},\ldots ,u_{n})})\) is determined by a stable equivalence \(T(f_{(u_{1},\ldots ,u_{n})})\sim R/(u_{1},\ldots ,u_{n})\). The Authors show how to interpret the topological Hochschild homology of Thom spectra in this setup.

Reviewer: Daniel Juan Pineda (Michoacan)

MSC:

19D55	\(K\)-theory and homology; cyclic homology and cohomology
55P43	Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
55N20	Generalized (extraordinary) homology and cohomology theories in algebraic topology
55P42	Stable homotopy theory, spectra
55P48	Loop space machines and operads in algebraic topology

Keywords:

Thom spectrum; structured ring spectrum; topological Hochschild homology