Footprints of geodesics in persistent homology (Q2148349)

The by now classical constructions of the Čech and Rips filtration in persistent homology mostly are based on the Euclidean distance. This paper focuses on persistent homology via Čech and Rips filtrations of geodesic spaces. The current work may lead to interesting connections between geometric features of geodesic spaces with algebraic elements of persistent homology. Particularly, the author studies persistent homology of geodesic circles in a geodesic surface under geodesic distance and explains how they may generate non-trivial odd-dimensional and two-dimensional algebraic elements, which provides a link between persistent homology and the length spectrum in Riemannian geometry. The topic of this paper is very interesting and offers unique insights towards connections between persistent homology and length geometry geodesic spaces.