Computation of cubical homology, cohomology, and (co)homological operations via chain contraction
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Computation of cubical homology, cohomology, and (co)homological operations via chain contraction Paweł Pilarczyk · Pedro Real
Received: 25 April 2013 / Accepted: 17 March 2014 © Springer Science+Business Media New York 2014
Abstract We introduce algorithms for the computation of homology, cohomology, and related operations on cubical cell complexes, using the technique based on a chain contraction from the original chain complex to a reduced one that represents its homology. This work is based on previous results for simplicial complexes, and uses Serre’s diagonalization for cubical cells. An implementation in C++ of the introduced algorithms is available at http://www.pawelpilarczyk.com/chaincon/ together with some examples. The paper is self-contained as much as possible, and is written at a very elementary level, so that basic knowledge of algebraic topology should be sufficient to follow it. Keywords Algorithm · Software · Homology · Cohomology · Computational homology · Cup product · Alexander-Whitney coproduct · Chain homotopy · Chain contraction · Cubical complex Mathematics Subject Classification (2010) 55N35 · 52B99 · 55U15 · 55U30 · 55-04 Communicated by: D. N. Arnold P. Pilarczyk () Universidade do Minho, Centro de Matem´atica, Campus de Gualtar, 4710-057 Braga, Portugal e-mail: [email protected] URL: http://www.pawelpilarczyk.com/ P. Real Departamento de Matem´atica Aplicada I, E. T. S. de Ingenier´ıa Inform´atica, Universidad de Sevilla, Avenida Reina Mercedes s/n, 41012 Sevilla, Spain e-mail: [email protected] URL: http://investigacion.us.es/sisius/sis showpub.php?idpers=1021 Present Address: P. Pilarczyk IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria
P. Pilarczyk, P. Real
1 Introduction A full cubical set in Rn is a finite union of n-dimensional boxes of fixed size (called cubes for short) aligned with a uniform rectangular grid in Rn . Due to the product structure and alignment with coordinate axes, using full cubical sets for approximating bounded subsets of Rn is very natural: a cube containing a point x = (x1 , . . . , xn) ∈ Rn can be instantly calculated by simply truncating the Cartesian coordinates of x down to the nearest grid thresholds. For simplicity of notation, these thresholds can be set to the integers, so that the cubes are of unitary size. Such cubical sets naturally correspond to 2D and 3D binary images. By analogy to simplicial complexes (see e.g. [44]), sets of cubes with their vertices, edges, faces, etc., yield a natural chain complex structure, which can be used to compute their homology groups. We refer to [34] for a comprehensive study of this subject, and to the [9] and the [10] for a representative implementation of homology computation algorithms focused specifically on cubical sets. Homology computation of cubical sets has already found some interesting applications. To mention a few of them, homology was used to extract topological information from medical images (e.g. [45]), to classify the complexity of patterns coming from numerical simulations of PDEs
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