Topological hochschild homology and cohomology of A ∞ ring spectra

Let A be an A ∞ 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 ∞ structure into THH(A), and allows us to study how THH(A) varies over the moduli space of A ∞ structures on A. As an example, we study how topological Hochschild cohomology of Morava K-theory varies over the moduli space of A ∞ 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 ∞ structure is "more commutative", topological Hochschild cohomology of Morava K-theory is some extension of Morava E-theory.