Unital associahedra and homotopy unital homotopy associative algebras

Unital associahedra and homotopy unital homotopy associative algebras

Mon, 05/03/2012
15:45 		Andy Tonks (London Metropolitan University) Topology Seminar Add to calendar L3
Unital associahedra and homotopy unital homotopy associative algebras
The classical associahedra are cell complexes, in fact polytopes, introduced by Stasheff to parametrize the multivariate operations naturally occurring on loop spaces of connected spaces. They form a topological operad $ Ass_\infty $ (which provides a resolution of the operad $ Ass $ governing spaces-with-associative-multiplication) and the complexes of cellular chains on the associahedra form a dg operad governing $ A_\infty $-algebras (that is, a resolution of the operad governing associative algebras). In classical applications it was not necessary to consider units for multiplication, or it was assumed units were strict. The introduction of non-strict units into the picture was considerably harder: Fukaya-Ono-Oh-Ohta introduced homotopy units for $ A_\infty $-algebras in their work on Lagrangian intersection Floer theory, and equivalent descriptions of the dg operad for homotopy unital $ A_\infty $-algebras have now been given, for example, by Lyubashenko and by Milles-Hirsch. In this talk we present the "missing link": a cellular topological operad $ uAss_\infty $ of "unital associahedra", providing a resolution for the operad governing topological monoids, such that the cellular chains on $ uAss_\infty $ is precisely the dg operad of Fukaya-Ono-Oh-Ohta. (joint work with Fernando Muro, arxiv:1110.1959, to appear Forum Math)