EVENT DETAILS AND ABSTRACT

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.