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

Abstract

Persistent homology typically studies the evolution of homology groups $H_p(X)$ (with coefficients in a field) along a filtration of topological spaces. $A_{\infty }$-persistence extends this theory by analysing the evolution of subspaces such as $V:=\text{Ker}{{\Delta }_n}_{|H_p(X)}\subseteq H_p(X)$, where $\{{\Delta }_m{\}}_{m\ge 1}$ denotes a structure of $A_{\infty }$-coalgebra on $H_{\ast }(X)$. In this paper we illustrate how $A_{\infty }$-persistence can be useful beyond persistent homology by discussing the topological meaning of $V$, which is the most basic form of $A_{\infty }$-persistence group. In addition, we explore how to choose $A_{\infty }$-coalgebras along a filtration to make the $A_{\infty }$-persistence groups carry more faithful information.

26 pages, 3 figures

References in corpus (2)

• Higher order Whitehead products and $L_{\infty }$ structures on the homology of a DGL
• A-infinity Persistence