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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An algorithmic approach to the construction
of homomorphisms induced by maps in homology


Authors: Madjid Allili and Tomasz Kaczynski
Journal: Trans. Amer. Math. Soc. 352 (2000), 2261-2281
MSC (1991): Primary 55-04; Secondary 54C60, 54H20, 05B25
Published electronically: November 18, 1999
MathSciNet review: 1694277
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Abstract: This paper is devoted to giving the theoretical background for an algorithm for computing homomorphisms induced by maps in homology. The principal idea is to insert the graph of a given continuous map $\, f \,$ into a graph of a multi-valued representable map $\, F$. The multi-valued representable maps have well developed continuity properties and admit a finite coding that permits treating them by combinatorial methods. We provide the construction of the homomorphism $\, F_* \,$ induced by $\, F \,$ such that $\, F_* = f_*$. The presented construction does not require subsequent barycentric subdivisions and simplicial approximations of $\, f$. The main motivation for this paper comes from the project of computing the Conley Index for discrete dynamical systems.


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Additional Information

Madjid Allili
Affiliation: Center for Dynamical Systems and Nonlinear Studies, School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Email: allili@math.gatech.edu

Tomasz Kaczynski
Affiliation: Département de Mathématiques et d’Informatique, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada
Email: kaczyn@dmi.usherb.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02527-1
PII: S 0002-9947(99)02527-1
Keywords: Algorithm, homology, representable map, Vietoris map
Received by editor(s): June 2, 1997
Received by editor(s) in revised form: January 14, 1998
Published electronically: November 18, 1999
Additional Notes: The second author was supported by grants from NSERC of Canada and FCAR of Quebec.
Article copyright: © Copyright 2000 American Mathematical Society