Contents

cohomology

# Contents

## Idea

In the context of persistent homology, the well group (EMP 11) of a continuous function $f$ into a metric space is a homological measure of how robust the level sets of $f$ are against deformations of $f$. Concretely, the Well group at radius $r$ 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 $f$ at most by $r$.

## Definition

Given a continuous function $f$ to a Euclidean space and a choice of topological subspace $A$ of the latter

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

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

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

(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