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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.

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
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[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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