Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

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

Author(s): Madjid Allili; Tomasz Kaczynski
Journal: Trans. Amer. Math. Soc. 352 (2000), 2261-2281.
MSC (1991): Primary 55-04; Secondary 54C60, 54H20, 05B25
Posted: November 18, 1999
MathSciNet review: 1694277
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

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.

References:

1.: M. Allili, Une approche algorithmique pour le calcul de l'homologie de fonctions continues, Ph.D. thesis, Département de mathématique et d'informatique, Université de Sherbrooke, January 1999.
2.: E. Begle, The Vietoris mapping theorem for bicompact spaces, Ann. Math., 51 (1950), 534-543. MR 11:677b
3.: C. A. Delfinado, H. Edelsbrunner, An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere, Computer Aided Geometric Design, 12 (1995), 771-784. MR 96h:55004
4.: T. K. Dey, S. Guha, Algorithms for manifolds and simplicial complexes in Euclidean 3-space, In Proc. 28th ACM Sympos. Theory Comput. (1996), 398-407. CMP 97:06
5.: T. K. Dey, H. Edelsbrunner, S. Guha, Computational Topology, Advances in discrete and computational geometry (South Hadley, MA, 1996), 109-143. CMP 99:05
6.: D. Dold, Lectures on algebraic topology, Springer-Verlag, Berlin-Heidelberg-New York, (1972). MR 96c:55001
7.: R. Ehrenborg, G. Hetyei, Generalizations of Baxter's Theorem and Cubical Homology, J. Combinatorial Theory, Series A, 69 (1995), 233-287. MR 96k:05041
8.: L. Górniewicz, Homological methods in fixed point theory of multi-valued map, Dissertationes Math. (Rozprawy Mat.) 129 (1976), 1-71. MR 52:15438
9.: C. S. Iliopoulos, Worst case complexity bounds on algorithms for computing the canonical structure of finite abelian groups and Hermite and Smith normal forms of an integer matrix, SIAM J. Computing 18 (1989), 658-669. MR 91a:20065
10.: T. Kaczynski, M. Mrozek, Conley index for discrete multi-valued dynamical systems, Topology Appl. 65 (1995), 83-96. MR 97d:54066
11.: -, Stable index pairs for discrete dynamical systems, Canad. Math. Bull. 40 (1997), 448-455. MR 99g:54038
12.: T. Kaczynski, M. Mrozek, M. \'{S}lusarek, Homology computation by reduction of chain complexes, Comput. Math. Appl. 33 (1998), 59-70. MR 99a:18001
13.: T. Kaczynski, H. Karloff, K. Mischaikow, M. Mrozek, Introduction to Algebraic Topology: A Computational Perspective, book in progress.
14.: A. Lecki, Z. Szafraniec, An algebraic method for calculating the topological degree, Banach Center Publ. Vol. 35, Warsaw 1996, 73-83. MR 98b:55002
15.: W. S. Massey, A basic course in algebraic topology, Springer-Verlag, New York (1991). MR 92c:55001
16.: K. Mischaikow, Conley Index Theory, Dynamical Systems: Montecatini Terme 1994, eds. L. Arnold and C. Jones, Lecture Notes in Math. No. 1609, Springer 1995. MR 97a:58109
17.: K. Mischaikow, M. Mrozek, A. Szymczak, J. Reiss, From time series to symbolic dynamics: An algebraic topological approach, preliminary version.
18.: K. Mischaikow, M. Mrozek, Chaos in the Lorenz equations: a computer assisted proof, Bull. Amer. Math. Soc. (N.S.) 32 (1995), 66-72. MR 95e:58121
19.: K. Mischaikow, M. Mrozek, A. Szymczak, Chaos in the Lorenz equations: a computer assisted proof, Part III: the classical case. Preprint.
20.: M. Mrozek, Topological invariants, multivalued maps and computer assisted proofs in dynamics, Computers and Mathematics 32 (1996), 83-104. MR 97h:58144
21.: J. R. Munkers, Elements of algebraic topology, Addison-Wesley (1984). MR 85m:55001
22.: H. W. Siegberg, G. Skordev, Fixed point index and chain approximations, Pacific Journal of Mathematics 102, No. 2 (1982), 455-486. MR 84d:55003
23.: A. Szymczak, A combinatorial procedure for finding isolating neighbourhoods and index pairs, Proc. of the Royal Soc. of Edinburgh 127 A (1997), 1075-1088. MR 98i:58151
24.: P. Zgliczynski, Fixed points for iterations of maps, topological horseshoe and chaos, Top. Meth. in Nonlin. Anal. 8 (1996), 169-177. MR 98m:58106
25.: -, Computer assisted proofs of chaos in the Rössler equations and the Hénon map, Nonlinearity 10 (1997), 243-252. MR 98g:58120

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|