On the Multiplicative Structure of Topological Hochschild Homology

We show that the topological Hochschild homology THH(R) of an En-ring spectrum R is an En−1-ring spectrum. The proof is based on the fact that the tensor product of the operad Ass for monoid structures and the the little n-cubes operad Cn is an En+1-operad, a result which is of independent interest. In 1993 Deligne asked whether the Hochschild cochain complex of an associative ring has a canonical action by the singular chains of the little 2-cubes operad. Affirmative answers for differential graded algebras in characteristic 0 have been found by Kontsevich and Soibelman [11], Tamarkin [15] and [16], and Voronov [18]. A more general proof, which also applies to associative ring spectra is due to McClure and Smith [14]. In [10] Kontsevich extended Deligne’s question: Does the Hochschild cochain complex of an En differential graded algebra carry a canonical En+1-structure? We consider the dual problem: Given a ring R with additional structure, how much structure does the topological Hochschild homology THH(R) of R inherit from R? The close connection of THH with algebraic K-theory and with structural questions in the category of spectra make multiplicative structures on THH desirable. In his early work on topological Hochschild homology of functors with smash product Bökstedt proved that THH of a commutative such functor is a commutative ring spectrum (unpublished). The discovery of associative, commutative and unital smash product functors of spectra simplified the definition of THH and the proof of the corresponding result for E∞-ring spectra considerably (e.g. see [13]).