Optimising the topological information of the $A_\infty$-persistence groups

Persistent homology typically studies the evolution of homology groups Hp(X) (with coefficients in a field) along a filtration of topological spaces. A∞-persistence extends this theory by analysing the evolution of subspaces such as V := Ker ∆n|Hp(X) ⊆ Hp(X), where {∆m}m≥1 denotes a structure of A∞-coalgebra on H∗(X). In this paper we illustrate how A∞-persistence can be useful beyond persistent homology by discussing the topological meaning of V , which is the most basic form of A∞-persistence group. In addition, we explore how to choose A∞-coalgebras along a filtration to make the A∞-persistence groups carry more faithful information.