an:07184351
Zbl 1443.55002
Virk, ??iga
1-Dimensional intrinsic persistence of geodesic spaces
EN
J. Topol. Anal. 12, No. 1, 169-207 (2020).
00448207
2020
j
55N31 57N65 55N05 55N35 55Q05 53C22 30F99 53B21
geodesic spaces; persistence; Rips filtrations; ??ech filtration; fundamental group; geodesic; homology base
In this paper the author develops the theory of 1-dimensional persistence, which refers to the persistence obtained from any filtration by applying the fundamental group or the homology group (with any coefficients) functor.
The main new results developed by this paper are roughly the following:
(1) a Rips-critical point of persistence corresponds to an isometrically embedded circle(s) of length \(3c\), which arises from the boundaries of critical triangles;
(2) 0 is the only possible accumulation point of the set of critical points, with the latter being finite for locally contractible spaces;
(3) persistence measures precisely the `size' of holes measured by the length (equivalently the diameter or the radius of the smallest enclosing disc) of the corresponding embedded circle;
(4) persistences via Rips and ??ech filtrations are isomorphic up to a factor 3/4.
Bo??ena Pi??tek (Gliwice)