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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.
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