Abstract: (Joint work with J. Rognes, Oslo)

In this talk I will explain the construction and properties of a certain chain complex H(t) associated to a given fixed twisting cochain t. Since this construction generalizes that of both the Hochschild complex of an associative algebra and the coHochschild complex of a coassociative coalgebra, we call H(t) the Hochschild complex of t.

I'll give conditions under which H(t) admits power maps extending the usual power maps on a Hopf algebra. In particular, it turns out that both the Hochschild complex of any cocommutative Hopf algebra and the coHochschild complex of the normalized chain complex of a double suspension admit power maps, which are algebraic models for the topological power maps on free loop spaces. This algebraic model of the topological power map is a crucial element of the construction of our model for computing spectrum homology of topological cyclic homology of spaces.