Topological Hochschild homology and cohomology of $A_\infty$ ring spectra

Abstract

Let $A$ be an $A_{\infty }$ ring spectrum. We use the description from our preprint [math.AT/0612165] of the cyclic bar and cobar construction to give a direct definition of topological Hochschild homology and cohomology of $A$ using the Stasheff associahedra and another family of polyhedra called cyclohedra. This construction builds the maps making up the $A_{\infty }$ structure into $THH\left(A\right)$, and allows us to study how $THH\left(A\right)$ varies over the moduli space of $A_{\infty }$ structures on $A$.

As an example, we study how topological Hochschild cohomology of Morava $K$–theory varies over the moduli space of $A_{\infty }$ structures and show that in the generic case, when a certain matrix describing the noncommutativity of the multiplication is invertible, topological Hochschild cohomology of $2$–periodic Morava $K$–theory is the corresponding Morava $E$–theory. If the $A_{\infty }$ structure is “more commutative”, topological Hochschild cohomology of Morava $K$–theory is some extension of Morava $E$–theory.