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.
Keywords
structured ring spectra, Morava K-theory, associahedra,
cyclohedra, topological Hochschild homology
