well group





Special and general types

Special notions


Extra structure





In the context of persistent homology, the well group (EMP 11) of a continuous function ff into a metric space is a homological measure of how robust the level sets of ff are against deformations of ff. Concretely, the Well group at radius rr of a given point in the codomain consists of all those homology-classes of the domain which are reflected in the level sets of every continuous function whose values differ from those of ff at most by rr.


Given a continuous function ff to a Euclidean space and a choice of topological subspace AA of the latter

f 1(A) X f A n \array{ f^{-1}(A) &\subset & X \\ \downarrow && \big\downarrow {}^{\mathrlap{f}} \\ A &\subset& \mathbb{R}^n }

the well groups at radius r(0,)r \in (0,\infty) are the intersections of the ordinary homology groups of the pre-images g 1(A)Xg^{-1}(A) \subset X for all continuous functions Xg nX \overset{g}{\to} \mathbb{R}^n whose maximal distance from ff is |gf|r\left \vert g-f\right \vert \leq r.

W (f,r)Xg n|gf|rimage(H (g 1(A))H (g 1(A)X)H (X)) W_\bullet(f,r) \;\coloneqq\; \underset{ {X \overset{g}{\to} \mathbb{R}^n} \atop {\left \vert g-f\right \vert \leq r} }{\bigcap} image \Big( H_\bullet\big( g^{-1}(A) \big) \overset{ H_\bullet\big( g^{-1}(A) \subset X \big) }{\longrightarrow} H_\bullet \big( X \big) \Big)

(e.g. Franek-Krčál 16, p. 2)


The concept of well groups was introduced in

  • Herbert Edelsbrunner, Dmitriy Morozov, Amit Patel, Quantifying Transversality by Measuring the Robustness of Intersections, Foundations of Computational Mathematics, 11(3):345–361, 2011 (arXiv:0911.2142)

  • Paul Bendich, Herbert Edelsbrunner, Dmitriy Morozov, Amit Patel, The Robustness of Level Sets, In: M. de Berg, U. Meyer (eds.) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6346. Springer (doi:10.1007/978-3-642-15775-2_1)

  • Paul Bendich, Herbert Edelsbrunner, Dmitriy Morozov, Amit Patel, Homology and Robustness of Level and Interlevel Sets, Homology, Homotopy and Applications, vol. 15, pages 51-72, 2013 (euclid:1383943667)

Review in:

  • Sara Kališnik, Section 4.2.2 of: Persistent homology and duality, 2013 (pdf, pdf)

Review, computational analysis and discussion of Cohomotopy as an improvement over homology well groups is in

Created on February 6, 2020 at 04:54:45. See the history of this page for a list of all contributions to it.