nLab well group

Contents

Context

Algebraic topology

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 Čech 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.

As the level varies, the collection of well groups form a zigzag persistence module, also called a well module (EMP 11, Sec. 3).

Since a well group becomes trivial as soon as one of the level sets is empty, the non-triviality of a well group proves that the existence of a non-empty level set of ff is robust within deformations of size <r\lt r. In topological data analysis this may be used to detect if there are guaranteed to be any data points at all meeting a certain target of indicator values, known with limited precision, see there for more.

But well groups are known not to resolve all relevant cases and are not known to be computable in all relevant cases (Franek & Krčál 2016). An enhancement of well groups (from homology to cohomotopy) which fixes these problems is persistent cohomotopy (Franek & Krčál 2017, 2018).

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 Čech 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)

Remark

Since the preimages g 1(A)g^{-1}(A) need not be CW-complexes, it is important to use Čech homology in the above definition. With singular homology the definition would trivialize (FK16, p. 3 and Sec. 2).

References

The concept of well groups was introduced in

Review in:

Review, computational analysis and discussion of (persistent) Cohomotopy as an improvement over homology well groups:

Survey:

Last revised on May 23, 2022 at 08:32:19. See the history of this page for a list of all contributions to it.