Homotopical decompositions of simplicial and Vietoris Rips complexes (Q2240094)

Decomposition is essential in computational topology and topological data analysis. What is the relationship between the composing parts and their union? Quoting the paper, ``under what circumstances is the inclusion \(K_X \cup K_Y \hookrightarrow K\) a weak equivalence, or homology isomorphism, or has highly connected homotopy fibers etc.?''. These kinds of questions, already investigated in [\textit{M. Adamaszek} et al., LIPIcs -- Leibniz Int. Proc. Inform. 99, Article 3, 15 p. (2018; Zbl 1489.68330); J. Appl. Comput. Topol. 4, No. 3, 425--454 (2020; Zbl 1455.55005)] are treated here in a very thorough way. Homotopy of simplicial sets, small categories and simplicial sets and complexes take three dense sections by which the paper is fully self-contained. The reader is then led gradually into the main theme: addition of one, two, several vertices take one section each. Two sections are dedicated to the key categorical tool: push-out. The paper turns then towards applications in the sections on clique complexes and Vietoris-Rips complexes; the latter subject is treated first for ``distances'' that are just reflexive and symmetric, then for pseudodistances in a separate section. A simple but meaningful example closes the paper. The strength of the paper is the detailed distinction of cases, each leading to a Hurewicz type of theorem. It has an admirable balance between general and particular, and is a very good example of application-oriented mathematics, where applications are an opportunity for developing theory.