Geometry and Topology 12:2 (2008) 987–1032

DOI: 10.2140/gt.2008.12.987

Abstract

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.

Keywords

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

Mathematical Subject Classification

Primary: 55P43

Secondary: 18D50, 55S35

References

Publication

Received: 5 April 2007
Accepted: 8 February 2008
Published: 12 May 2008
Proposed: Bill Dwyer
Seconded: Paul Goerss, Ralph Cohen

Authors