Associahedra, Cyclohedra and a Topological solution to the A_{\infty}--Deligne conjecture

TL;DR

This paper provides a topological approach to the $ ext{A}_ ext{infty}$ Deligne conjecture using associahedra and cyclohedra, constructing CW complexes that model the little discs operad and act on Hochschild cochains.

Contribution

It introduces explicit realizations of associahedra and cyclohedra via trees, constructs CW complexes modeling the little discs, and offers a minimal topological solution to the $ ext{A}_ ext{infty}$ Deligne conjecture.

Findings

Constructed CW complexes with cells indexed by products of polytopes.

Provided new decompositions of cyclohedra into products of cubes and simplices.

Demonstrated that the cellular chains act on Hochschild cochains of $ ext{A}_ ext{infty}$-algebras.

Abstract

We give a topological solution to the $\Ainf$ Deligne conjecture using associahedra and cyclohedra. For this 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 acts on the Hochschild cochains of an $\Ainf$ algebra yielding an explicit, topological and minimal solution to the $\Ainf$ Deligne conjecture. Along the way we obtain new results about the cyclohedra, such as a new decompositions into products of cubes and simplices, which can be used to realize them via a new iterated blow--up construction.