Topological Hochschild Homology of K/p as a K_p^\wedge module

2010/06/22 by Samik Basu, Basu, Samik
Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT

paper · pdf · doi:10.48550/arxiv.1006.4347

arxiv created 2010/12/03 · arxiv updated 2012/03/27

Abstract

Let R be an E_∞-ring spectrum. Given a map ζ from a space X to BGL₁R, one can construct a Thom spectrum, X^ζ, which generalises the classical notion of Thom spectrum for spherical fibrations in the case R=S⁰, the sphere spectrum. If X is a loop space (≃ ΩY) and ζ is homotopy equivalent to Ωf for a map f from Y to B²GL₁R, then the Thom spectrum has an A_∞-ring structure. The Topological Hochschild Homology of these A_∞-ring spectra is equivalent to the Thom spectrum of a map out of the free loop space of Y. This paper considers the case X=S¹, R=K_p^\wedge, the p-adic K-theory spectrum, and ζ= 1-p ∈ π₁BGL₁K_p^\wedge. The associated Thom spectrum (S¹)^ζ is equivalent to the mod p K-theory spectrum K/p. The map ζ is homotopy equivalent to a loop map, so the Thom spectrum has an A_∞-ring structure. I will compute π_*THH^{K_p^\wedge}(K/p) using its description as a Thom spectrum.

Related

• THH of Thom spectra that are E_∞ ring spectra
• Topological Hochschild Homology of H(Z/p^k)
• Topological Hochschild homology of X(n)
• Topological Hochschild homology of the image of j
• E_∞ ring spectra and elements of Hopf invariant 1