Geometry & Topology

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

Vigleik Angeltveit

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.

Article information

Source
Geom. Topol., Volume 12, Number 2 (2008), 987-1032.

Dates
Accepted: 8 February 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800067

Digital Object Identifier
doi:10.2140/gt.2008.12.987

Mathematical Reviews number (MathSciNet)
MR2403804

Zentralblatt MATH identifier
1149.55006

Citation

Angeltveit, Vigleik. Topological Hochschild homology and cohomology of $A_\infty$ ring spectra. Geom. Topol. 12 (2008), no. 2, 987--1032. doi:10.2140/gt.2008.12.987. https://projecteuclid.org/euclid.gt/1513800067

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