Abstract view
Ideas from Zariski Topology in the Study of Cubical Homology


Published:20071001
Printed: Oct 2007
Tomasz Kaczynski
Marian Mrozek
Anik Trahan
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 VietorisBegle 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: 
5504, 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] Setvalued maps [See also 26E25, 28B20, 47H04, 58C06] Nonnumerical algorithms Symbolic computation and algebraic computation [See also 11Yxx, 12Y05, 13Pxx, 14Qxx, 16Z05, 1708, 33F10] Image processing
5504  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  Setvalued maps [See also 26E25, 28B20, 47H04, 58C06] 68W05  Nonnumerical algorithms 68W30  Symbolic computation and algebraic computation [See also 11Yxx, 12Y05, 13Pxx, 14Qxx, 16Z05, 1708, 33F10] 68U10  Image processing
