The homological hexagonal lemma
type: Item
instance of: scholarly article
title: The homological hexagonal lemma
zbMATH Open document ID: 1409.13053
author: Francis Sergeraert
published in: Georgian Mathematical Journal
publication date: 2019-01-11T00:00:00Z
review text: The author proposes an approach to constructive homological algebra based on the homological perturbation theorem, reworked with what he calls the homological hexagonal lemma. The perturbation theorem, which originated in an unpublished work of Michael Barratt, has undergone several ``renovations'' over the last half-century. The present approach reduces this theorem to Gaussian elimination and achieves a remarkable reduction of computational complexity of homology groups of complexes in a variety of contexts. One of them deals with discrete vector fields, as defined by \textit{R. Forman} [Adv. Math. 134, No. 1, 90--145 (1998; Zbl 0896.57023)]. As the author points out, this allows ``\dots a totally new understanding of the Eilenberg-Zilber theorem \dots''. \par This interesting paper is written in a less formal way, which should be viewed as an advantage, although at some places the reviewer was puzzled by (non-mathematical) enigmatic sentences. As another advantage of this paper, we mention applications in less algebraic contexts like Banach spaces or even Fréchet spaces.
reviewed by: Alex Martsinkovsky
Mathematics Subject Classification ID: 13P20
Mathematics Subject Classification ID: 18G35
Mathematics Subject Classification ID: 57N60
zbMATH DE Number: 7000277
zbMATH Keywords: homological hexagonal lemma
zbMATH Keywords: homological perturbation theorem
zbMATH Keywords: discrete vector field
MaRDI profile type: Publication
Wikidata QID: Q124809214
full work available at URL: gmj-2018-0055
OpenAlex ID: W2891745946
cites work: On the groups \(H(\Pi,n)\). I
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cites work: Q5550390
cites work: Programming before theorizing, a case study
cites work: Constructive algebraic topology
cites work: Un théorème de fonctions implicites sur certains espaces de Fréchet et quelques applications
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DOI: 10.1515/GMJ-2018-0055
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description: scientific article; zbMATH DE number 7000277