Associahedra, cyclohedra and a topological solution to the A ∞ Deligne conjecture

Associahedra, cyclohedra and a topological solution to the A ∞ Deligne conjecture

We give a topological solution to the A ∞ Deligne conjecture using associahedra and cyclohedra. To this end, we construct three CW complexes whose cells are indexed by products of polytopes. Giving new explicit realizations of the polytopes in terms of different types of trees, we are able to show t...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) Vol. 223; no. 6; pp. 2166 - 2199
Main Authors: Kaufmann, Ralph M., Schwell, R.
Format: Journal Article
Language:English
Published: Elsevier Inc 01-04-2010
Subjects:
Actions on Hochschild complexes
Associahedra
Cyclohedra
Deligne's conjecture
Homotopy algebras
Little discs
Operads
Polytopes
Polytopes
Little discs
Actions on Hochschild complexes
Deligne's conjecture
Homotopy algebras
Associahedra
Operads
Cyclohedra
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Summary:We give a topological solution to the A ∞ Deligne conjecture using associahedra and cyclohedra. To this end, we construct three CW complexes whose cells are indexed by products of polytopes. Giving new explicit realizations of the polytopes in terms of different types of trees, we are able to show that the CW complexes are cell models for the little discs. The cellular chains of one complex in particular, which is built out of associahedra and cyclohedra, naturally act on the Hochschild cochains of an A ∞ algebra yielding an explicit, topological and minimal solution to the A ∞ Deligne conjecture. Along the way we obtain new results about the cyclohedra, such as new decompositions into products of cubes and simplices, which can be used to realize them via a new iterated blow-up construction.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2009.11.001