nLab well group

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 Čech 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 $r$.

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 $f$ is robust within deformations of size $\lt r$. 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 $f$ 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 Čech 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)

Remark

Since the preimages $g^{-1}(A)$ 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).

Related concepts

• persistence module

• zigzag persistence

• stability of persistence diagrams

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 (2011) 345–361 $[$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 6346 Springer (2010) $[$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, 15 (2013) 51-72 $[$euclid:1383943667$]$

Review in:

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

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

• Peter Franek, Marek Krčál, On Computability and Triviality of Well Groups, Discrete Comput Geom (2016) 56: 126 (arXiv:1501.03641, doi:10.1007/s00454-016-9794-2)

• Peter Franek, Marek Krčál, Persistence of Zero Sets, Homology, Homotopy and Applications, Volume 19 (2017) Number 2 (arXiv:1507.04310, doi:10.4310/HHA.2017.v19.n2.a16)

• Peter Franek, Marek Krčál, Hubert Wagner, Solving equations and optimization problems with uncertainty, J Appl. and Comput. Topology (2018) 1: 297 (arxiv:1607.06344, doi:10.1007/s41468-017-0009-6)

Survey:

• Peter Franek, Marek Krčál, Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory, talk at ACAT meeting 2015 (pfd slides)