The homological hexagonal lemma
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 The perturbation theorem, which originated in an unpublished work of Michael Barratt, has undergone several ``renovations'' over the last half-century.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.
scientific article; zbMATH DE number 7000277
scientific article