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

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
Full Text: DOI
[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
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