Let be an 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 using the Stasheff associahedra and another family of polyhedra called cyclohedra. This construction builds the maps making up the structure into , and allows us to study how varies over the moduli space of structures on .
As an example, we study how topological Hochschild cohomology of Morava –theory varies over the moduli space of structures and show that in the generic case, when a certain matrix describing the noncommutativity of the multiplication is invertible, topological Hochschild cohomology of –periodic Morava –theory is the corresponding Morava –theory. If the structure is “more commutative”, topological Hochschild cohomology of Morava –theory is some extension of Morava –theory.
"Topological Hochschild homology and cohomology of $A_\infty$ ring spectra." Geom. Topol. 12 (2) 987 - 1032, 2008. https://doi.org/10.2140/gt.2008.12.987