Families of $A_{\infty }$ algebras and homotopy group actions

Abstract

We define homotopy group actions in terms of families of $A_{\infty }$ algebras indexed by a manifold M. We give explicit formulae for the $A_{\infty }$ morphism induced by a path on the manifold and for the $A_{\infty }$ homotopy corresponding to a pair of homotopic paths. Finally, we compute examples for finite groups and finitely generated free nonabelian groups and determine that every homotopy group action by a finite group is homotopic to a strict group action.

23 pages, 4 figures