# zbMATH — the first resource for mathematics

1-Dimensional intrinsic persistence of geodesic spaces. (English) Zbl 1443.55002
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 Čech filtrations are isomorphic up to a factor 3/4.

##### MSC:
 55N31 Persistent homology and applications, topological data analysis 57N65 Algebraic topology of manifolds 55N05 Čech types 55N35 Other homology theories in algebraic topology 55Q05 Homotopy groups, general; sets of homotopy classes 53C22 Geodesics in global differential geometry 30F99 Riemann surfaces 53B21 Methods of local Riemannian geometry
Full Text:
##### References:
 [1] Adamaszek, M. and Adams, H., The Vietoris-Rips complexes of a circle, Pacific J. Math.290 (2017) 1-40. · Zbl 1366.05124 [2] M. Adamaszek, H. Adams and F. Frick, Metric reconstruction via optimal transport, preprint (2017), arXiv:1706.04876. · Zbl 1406.53045 [3] Brazas, J. and Fabel, P., Thick Spanier groups and the first shape group, Rocky Mountain J. Math.44 (2014) 1415-1444. · Zbl 1306.57004 [4] N. Brodskiy, J. Dydak, B. Labuz and A. Mitra, Topological and uniform structures on universal covering spaces, preprint (2012), arXiv:1206.0071. · Zbl 1260.55013 [5] Cencelj, M., Dydak, J., Vavpetič, A. and Virk, Ž., A combinatorial approach to coarse geometry, Topology Appl.159 (2012) 646-658. · Zbl 1273.54030 [6] Chazal, F., Crawley-Boevey, W. and de Silva, V., The observable structure of persistence modules, Homol. Homotopy Appl.18 (2016) 247-265. · Zbl 1377.55015 [7] Chazal, F., de Silva, V., Glisse, M. and Oudot, S., The Structure and Stability of Persistence Modules, (Springer, 2016). · Zbl 1362.55002 [8] Chazal, F., de Silva, V. and Oudot, S., Persistence stability for geometric complexes, Geom. Dedicata173 (2014) 193. · Zbl 1320.55003 [9] T. K. Dey and K. Li, Topology from data via geodesic complexes, http://web.cse.ohio-state.edu/∼dey.8/paper/geodesic/geocom.pdf. [10] Edelsbrunner, H. and Harer, J. L., Computational Topology: An Introduction (Amer. Math. Soc., 2010). · Zbl 1193.55001 [11] Edelsbrunner, H. and Pausinger, F., Approximation and convergence of the intrinsic volume, Adv. Math.287 (2016) 674-703. · Zbl 1328.65063 [12] Fischer, H., Repovš, D., Virk, Ž. and Zastrow, A., On semilocally simply connected spaces, Topology Appl.158 (2011) 397-408. · Zbl 1219.54028 [13] E. Gasparovic, M. Gommel, E. Purvine, R. Sazdanovic, B. Wang, Y Wang and L. Ziegelmeier, A complete characterization of the $$1$$-dimensional intrinsic Čech persistence diagrams for metric graphs, preprint (2017), arXiv:1702.07379 [math.AT]. · Zbl 1422.55037 [14] Hatcher, A., Algebraic Topology (Cambridge Univ. Press, 2002). · Zbl 1044.55001 [15] Letscher, D., On persistent homotopy, knotted complexes and the Alexander module, in Proc. ITCS ’12 Proc. 3rd Innovations in Theoretical Computer Science Conf. (Cambridge, MA, USA, 2012), pp. 428-441. · Zbl 1348.57026 [16] Niyogi, P., Smale, S. and Weinberger, S., Finding the homology of submanifolds with high confidence from random samples, Discrete Comput. Geom.39 (2008) 419-441. · Zbl 1148.68048 [17] Ž. Virk, Approximations of $$1$$-dimensional intrinsic persistence of geodesic spaces and their stability, preprint (2017), arXiv:1711.05111. · Zbl 1412.55018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.