## EVENT DETAILS AND ABSTRACT

**Topology Seminar**
**Title:** Families of A-infinity algebras and homotopy group actions

**Speaker:** Emma Smith Zbarsky

**Speaker Info:** U Chicago

**Brief Description:**

**Special Note**: **Different Day of the Week.**

**Abstract:**

I will begin by defining families of A-infinity algebras indexed by a manifold M as solutions to the Maurer-Cartan equation on the de Rham complex of M with values in a particular locally trivial sheaf of Lie algebras on M built out of the Hochschild cochain complex of the underlying vector space of the A-infinity algebra. This will lead us to an explicit formulation for the A-infinity morphism F_{x -> y}:A_x -> A_y for x,y path connected points of M. Such an F is path-dependent. After showing that differential homotopies correspond to classical homotopies I will prove an explicit form for a differential homotopy
relating a family of homotopies F_t:A_x -> A_y as before. I shall close by computing the cohomology of the total complex defined above when M is a K(G,1) for G finite or finitely generated free nonabelian. Computation shows that in these cases the spectral sequence collapses at E^2. Thus, for a finite group G, every homotopy G action on an A-infinity algebra A has class representatives F_g:A -> A for all g in G which comprise a strict action.

**Date:** Thursday, May 07, 2009

**Time:** 4:10pm

**Where:** Lunt 104

**Contact Person:** Prof. Paul Goerss

**Contact email:** pgoerss@math.northwestern.edu

**Contact Phone:** 847-491-8544

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