Morse theory for cell complexes
type: Item
instance of: scholarly article
title: Morse theory for cell complexes
zbMATH Open document ID: 0896.57023
author: Robin Forman
published in: Advances in Mathematics
publication date: 1998-09-22T00:00:00Z
review text: This paper provides a discrete Morse theory for CW-complexes. The author proves an analogue of the main theorems of the classical theory and gets in the context of PL manifolds an interesting formula of the PL Poincaré conjecture. ``The following are equivalent: (1) (PL Poincaré conjecture) Let \(M\) be a PL \(n\)-manifold which is a homotopy \(n\)-sphere. Then \(M\) is a PL \(n\)-sphere; (2) Let \(M\) be a PL \(n\)-manifold which is a homotopy \(n\)-sphere. Then, by a series of bisections, \(M\) can be subdivided to a complex which has a Morse function with exactly 2 critical points.'' The author builds also the gradient vector field of a discrete (or combinatorial) Morse function and an associated differential complex with the same homology as the underlying manifold. This discrete Morse theory can then be used to give a Morse theoretic proof à la Milnor of the PL \(s\)-cobordism theorem [\textit{J. W. Milnor}, Lectures on the \(h\)-cobordism theorem (1965; Zbl 0161.20302)]. Main of the final part of the paper is devoted to characterize gradient vector fields of discete Morse functions. Doing that, the author shows that every discrete Morse function can be replaced by a self-indexing one, with the same critical points. In spite of some misprints or change without notice (e.g. combinatorial for discrete) this paper is pleasant and not too hard to read; there are many examples. It should become a reference in the subject.
Mathematics Subject Classification ID: 57R70
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 57R80
Mathematics Subject Classification ID: 57R60
Mathematics Subject Classification ID: 57Q99
zbMATH DE Number: 1146304
zbMATH Keywords: Morse theory
zbMATH Keywords: CW-complexes
zbMATH Keywords: PL Poincaré conjecture
zbMATH Keywords: discrete Morse function
reviewed by: Raymond Barre
MaRDI profile type: Publication
OpenAlex ID: W2064020260
cites work: Critical points and curvature for embedded polyhedra
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cites work: A combinatorial decomposition of simplicial complexes
cites work: Manifolds which are like projective planes
cites work: Determinants of Laplacians on graphs
cites work: Witten-Morse theory for cell complexes
cites work: Q5688873
cites work: Combinatorial vector fields and dynamical systems
cites work: Q5614468
cites work: Q4039743
cites work: Singularities of piecewise linear mappings. I Mappings into the real line
cites work: Q4027296
cites work: Q5607712
cites work: Differential topology from the point of view of simple homotopy theory
cites work: Q5543303
cites work: Q5542626
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cites work: Generalized Poincaré's conjecture in dimensions greater than four
cites work: Lectures on polyhedral topology
cites work: A combinatorial decomposition of acyclic simplicial complexes
cites work: Simplicial Spaces, Nuclei and m -Groups
cites work: Supersymmetry and Morse theory
DOI: 10.1006/AIMA.1997.1650
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description: scientific article; zbMATH DE number 1146304