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

Vigleik Angeltveit

Geometry & Topology 12 (2008) 987–1032

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.

structured ring spectra, Morava K-theory, associahedra, cyclohedra, topological Hochschild homology
Mathematical Subject Classification 2000
Primary: 55P43
Secondary: 18D50, 55S35
Received: 5 April 2007
Accepted: 8 February 2008
Published: 12 May 2008
Proposed: Bill Dwyer
Seconded: Paul Goerss, Ralph Cohen
Vigleik Angeltveit
University of Chicago
Department of Mathematics
5734 S University Ave
Chicago IL 60637