Optimising the topological information of the A_∞-persistence groups

06/19/2017

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by   Francisco Belchí, et al.

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Persistent homology typically studies the evolution of homology groups H_p(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 Δ_n_| H_p(X)⊆ H_p(X), where {Δ_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.