\(A_\infty\)-persistence. (English) Zbl 1330.55008
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.
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.
Reviewer: Jonathan Hodgson (Swarthmore)
MSC:
| 55N35 | Other homology theories in algebraic topology |
| 18D99 | Categorical structures |
| 57T99 | Homology and homotopy of topological groups and related structures |
Keywords:
persistent homology; \(A_{\infty}\)-persistence; \(A_{\infty}\)-coalgebra; applied algebraic topologyReferences:
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