Topological Hochschild homology and cohomology of A∞ ring spectra

Angeltveit, Vigleik

doi:10.2140/gt.2008.12.987

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

Vigleik Angeltveit

Geometry & Topology 12 (2008) 987–1032

DOI: 10.2140/gt.2008.12.987

Abstract

Let $A$ be an $A_{\infty}$ 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 $A_{\infty}$ 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.

Keywords

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

Mathematical Subject Classification 2000

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

Vigleik Angeltveit
University of Chicago Department of Mathematics 5734 S University Ave Chicago IL 60637 USA
http://www.math.uchicago.edu/~vigleik

Volume 12, issue 2 (2008)