Topological Hochschild homology and cohomology of A_\infty ring spectra

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

2006-12-06

arxiv.org/abs/math/0612164v1

Mathematics

Algebraic Topology

38 pages, 2 figures

Scientific paper

Let A be an A_\infty ring spectrum. We use the description from [2] 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(A), and allows us to study how THH(A) 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.

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