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

TL;DR

This paper introduces a framework for homotopy group actions using families of $A_ abla_$ algebras parametrized by manifolds, providing explicit formulas and examples, and showing finite group actions are homotopic to strict actions.

Contribution

It develops a new approach to homotopy group actions via $A_ abla_$ algebra families, with explicit morphism formulas and classification results for finite groups.

Findings

Explicit formulas for $A_ abla_$ morphisms induced by paths.

Homotopy group actions by finite groups are homotopic to strict actions.

Examples computed for finite and free nonabelian groups.

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.