\(A_\infty\)-persistence (Q2352516)

This paper introduces the notion of \(A_{\infty}\)-persistence in the homology of a filtration of a topological space. An \(A_{\infty}\)-coalgebra structure on a graded module \(M\) is given by a sequence of morphisms: \[ \Delta_{n} : M \longrightarrow M ^{\otimes n} \] of degree \(n-2\) satisfying certain relationships. In particular \(\Delta_{1}\) should be a differential on \(M\). This structure for the homology of a topological space carries information beyond that carried by the Betti numbers. Given a finite sequence of maps between topological spaces \[ K_{0} \rightarrow K_{1} \rightarrow \ldots \rightarrow K_{N} \] the authors define the \(p\)th \(\Delta_{n}\)-persistent groups between \(K_{i}\) and \(K_{j}\). The authors show how to assign to the filtration a canonical code of bars which effectively computes its \(\Delta_{n}\)-persistence.