## 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.

Original language | English |
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Pages (from-to) | 987-1032 |

Number of pages | 46 |

Journal | Geometry and Topology |

Volume | 12 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2008 |

Externally published | Yes |