On the definition of homological critical value

TL;DR

This paper examines issues with the standard definition of homological critical value, presents counterexamples showing the failure of the critical value lemma under that definition, and proposes a revised definition ensuring the lemma's validity.

Contribution

It identifies problems with the existing definition of homological critical value, offers a preferred alternative, and proves the critical value lemma holds under the new definition.

Findings

Counterexamples demonstrate failure of the critical value lemma under the original definition

A revised definition by Bubenik and Scott is proposed as more suitable

A modified version of the critical value lemma remains valid with the original definition

Abstract

We point out that there is a problem with the definition of homological critical value (as defined in the widely cited paper \cite{stability} by Cohen-Steiner, Edelsbrunner and Harer). Under that definition, the critical value lemma of \cite{stability} in fact fails. We provide several counterexamples and a definition (due to Bubenik and Scott \cite{categorification}) we feel should be preferred and under which the critical value lemma does indeed hold. One of the counterexamples we have found is a height function on a compact smooth manifold. In the end we prove that, despite all this, a modified version of the critical value lemma remains valid under the original definition.