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Ideas from Zariski Topology in the Study of Cubical Homology

  Published:2007-10-01
 Printed: Oct 2007
  • Tomasz Kaczynski
  • Marian Mrozek
  • Anik Trahan
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Abstract

Cubical sets and their homology have been used in dynamical systems as well as in digital imaging. We take a fresh look at this topic, following Zariski ideas from algebraic geometry. The cubical topology is defined to be a topology in $\R^d$ in which a set is closed if and only if it is cubical. This concept is a convenient frame for describing a variety of important features of cubical sets. Separation axioms which, in general, are not satisfied here, characterize exactly those pairs of points which we want to distinguish. The noetherian property guarantees the correctness of the algorithms. Moreover, maps between cubical sets which are continuous and closed with respect to the cubical topology are precisely those for whom the homology map can be defined and computed without grid subdivisions. A combinatorial version of the Vietoris-Begle theorem is derived. This theorem plays the central role in an algorithm computing homology of maps which are continuous with respect to the Euclidean topology.
MSC Classifications: 55-04, 52B05, 54C60, 68W05, 68W30, 68U10 show english descriptions Explicit machine computation and programs (not the theory of computation or programming)
Combinatorial properties (number of faces, shortest paths, etc.) [See also 05Cxx]
Set-valued maps [See also 26E25, 28B20, 47H04, 58C06]
Nonnumerical algorithms
Symbolic computation and algebraic computation [See also 11Yxx, 12Y05, 13Pxx, 14Qxx, 16Z05, 17-08, 33F10]
Image processing
55-04 - Explicit machine computation and programs (not the theory of computation or programming)
52B05 - Combinatorial properties (number of faces, shortest paths, etc.) [See also 05Cxx]
54C60 - Set-valued maps [See also 26E25, 28B20, 47H04, 58C06]
68W05 - Nonnumerical algorithms
68W30 - Symbolic computation and algebraic computation [See also 11Yxx, 12Y05, 13Pxx, 14Qxx, 16Z05, 17-08, 33F10]
68U10 - Image processing
 

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