The spectra ko and ku are not Thom spectra: an approach using THH

We apply an announced result of Blumberg-Cohen-Schlichtkrull to reprove (under restricted hypotheses) a theorem of Mahowald: the connective real and complex K-theory spectra are not Thom spectra. The construction of various bordism theories as Thom spectra served as a motivating example for the development of highly structured ring spectra. Various other examples of Thom spectra followed; for instance, various Eilenberg-Maclane spectra are known to be constructed in this way [Mah79]. However, Mahowald proved that the connective K-theory spectra ko and ku are not the 2-local Thom spectra of any vector bundles, and that the spectrum ko is not the Thom spectrum of a spherical fibration classified by a map of H-spaces [Mah87]. Rudyak later proved that ko and ku are not Thom spectra p-locally at odd primes p [Rud98]. There has been a recent clarification of the relationship between Thom spectra and topological Hochschild homology. Let BF be the classifying space for stable spherical fibrations. Theorem (Blumberg-Cohen-Schlichtkrull [BCS]). If Tf is a spectrum which is the Thom spectrum of a 3-fold loop map f : X → BF , then there is an equivalence THH(Tf) ≃ Tf ∧BX+.