A method for visualizing cohomology aspects of 3D objects

J.M. Berrio
jmberrio@sadiel.es
Applied Mathematics Dept., CS
Seville University
Avda. Reina Mercedes s/n
Spain

R. Gonzalez-Diaz
rododi@us.es
Applied Mathematics Dept., CS
Seville University
Avda. Reina Mercedes s/n
Spain

F. Leal
wes_mont@teleline.es
Applied Mathematics Dept., CS
Seville University
Avda. Reina Mercedes s/n
Spain

M.M. Lopez-Maraver
jmberrio@sadiel.es
Applied Mathematics Dept., CS
Seville University
Avda. Reina Mercedes s/n
Spain

P. Real
real@cica.es
Applied Mathematics Dept., CS
Seville University
Avda. Reina Mercedes s/n
Spain

Abstract

We propose here a method for visualizing homology, algebra cohomology, cohomology operations and cup-i products of 3D objects.
First of all, we give simplicial-based representations of three dimensional objects, essentially using a suitable tetrahedral decomposition of a cube. In order to compute the homology and cohomology of a given object, the first step in our method is the topological thinning of a simplicial complex which represents this object. From the resulting skeleton, the second step is to compute the algebraic invariants mentioned above.
From the point of view of the implementation, the graphical user interface of this method is presented in this paper and some examples concerning topological thinning and computation of the cochain operations cup-i products (see [1]) are discussed.

[1] J.R. Munkres. Elements of Algebraic Topology, Addison-Wesley Publishing Company, 1984.


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