On the definition of the homological critical value. (English) Zbl 1337.55009

Govc, Dejan

J. Homotopy Relat. Struct. 11, No. 1, 143-151 (2016).

Summary: We point out that there is a problem with the definition of the homological critical value as defined by D. Cohen-Steiner, H. Edelsbrunner and J. Harer in their seminal paper on stability of persistence diagrams [Discrete Comput. Geom. 37, No. 1, 103–120 (2007; Zbl 1117.54027)]. Under that definition, their critical value lemma in fact fails. We provide two counterexamples and a definition (due to Bubenik and Scott) we feel should be preferred. We also prove a version of the critical value lemma that remains valid under the original definition.

MSC:

55N99	Homology and cohomology theories in algebraic topology
54G20	Counterexamples in general topology
54E45	Compact (locally compact) metric spaces
68W30	Symbolic computation and algebraic computation

Keywords:

persistent homology; homological critical value; definition; counterexample

References:

[1]	Bubenik, P., Scott, J.A.: Categorification of persistent homology (2012). arXiv:1205.3669v1 · Zbl 1295.55005
[2]	Cohen-Steiner, D; Edelsbrunner, H; Harer, J, Stability of persistence diagrams, Discret. Comput. Geom., 37, 103-120, (2007) · Zbl 1117.54027
[3]	Govc, D.: On the definition of homological critical value (2013). arXiv:1301.6817v1
[4]	Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002) · Zbl 1044.55001
[5]	Lee, J.M.: Introduction to Smooth Manifolds. Springer, New York (2003)
[6]	Munkres, J.: Topology, a First Course. Prentice Hall, NJ (1975) · Zbl 0306.54001