nLab
well group

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

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.

Definition

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)

References

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.