Computing homology
type: Item
instance of: scholarly article
title: Computing homology
zbMATH Open document ID: 1028.55001
author: K. Mischaikow
author: Marian Mrozek
author: Tomasz Kaczynski
published in: Homology, Homotopy and Applications
publication date: 2004-01-13T00:00:00Z
full work available at URL: https://eudml.org/doc/50484
full work available at URL: volume5-2.htm
review text: This paper describes a computational homology based on the use of cubical complexes. The resulting algorithm is of order \(N^{3}\) in the maximum number of cubes in any dimension, but the authors claim that in many cases it will be faster than this. The basis of the algorithm lies in a collapsing technique that is used to reduce the number of cubes in each dimension while leaving the homology unchanged. More interesting is the computation of the homology of a map. Since the authors reject the use of simplices as leading to high dimensional chain groups they need to find an alternative to simplicial approximation. They do this by defining a cubical multi-valued map which maps points of the domain to subsets of the range (essentially the smallest cube with integer coordinates containing the image of the point; scaling makes this work) such that the image of each point in the domain is a cubical set. These maps permit the use of acyclic carriers and thus allow one to compute the homology of a map. Some of the algorithms, implemented in C++, are available at \url{http://www.math.gatech.edu/~chom}.
Mathematics Subject Classification ID: 55-04
Mathematics Subject Classification ID: 52B05
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 65D17
zbMATH DE Number: 1959268
zbMATH Keywords: homology computation
zbMATH Keywords: acyclic carrier
zbMATH Keywords: cubical complex
zbMATH Keywords: multivalued map
zbMATH Keywords: representable set
reviewed by: Jonathan Hodgson
MaRDI profile type: Publication
DOI: 10.4310/HHA.2003.V5.N2.A8
description: scientific article; zbMATH DE number 1959268