Fixed point index and chain approximations

Siegberg, Hans-Willi; Skordev, G.

doi:10.2140/pjm.1982.102.455

Cited by 31 publications

(33 citation statements)

References 23 publications

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“…We prove in Section 4 that our definition agrees with the classical definition of F * based on the Vietoris-Begle Theorem [2,22] and hence it coincides with f * : H(X) −→ H(Y ). Thus the main results of this paper are described by the following theorems: Theorem 3.1.…”

Section: Introductionmentioning

confidence: 54%

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

Állili

Kaczyński

1999

Trans. Amer. Math. Soc.

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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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Section: Introductionmentioning

confidence: 54%

“…The idea of our construction originates from [22]; however, the important asset is that, in our case, there are no barycentric subdivisions and subsequent chain-approximations required.…”

Section: Introductionmentioning

confidence: 99%

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

Állili

Kaczyński

1999

Trans. Amer. Math. Soc.

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“…A variety of concepts have been used to extend this result [6,7,8,9,21,27]. Recently, the author [13] used the dog-chases-rabbit principle to prove that every deformation of a uniquely arcwise connected continuum has a fixed point.…”

Section: Introductionmentioning

confidence: 99%

Fixed Points of Arc-Component-Preserving Maps

Hagopian¹

1988

Transactions of the American Mathematical Society

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ABSTRACT. The following classical problem remains unsolved:If M is a plane continuum that does not separate the plane and / is a map of M into M, must / have a fixed point?We prove that the answer is yes if / maps each arc-component of M into itself. Since every deformation of a space preserves its arc-components, this result establishes the fixed-point property for deformations of nonseparating plane continua. It also generalizes the author's theorem [10] that every arcwise connected nonseparating plane continuum has the fixed-point property. Our proof shows that every arc-component-preserving map of an indecomposable plane continuum has a fixed point. We also prove that every tree-like continuum that does not contain uncountably many disjoint triods has the fixed-point property for arc-component-preserving maps.

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“…The same technique was the main tool for developing the fixed point index with all properties (including commutativity and mod-/7-ρroperty multiplicity is proved in [26]) for multivalued maps of ANR's ( [9,11,25]). The main result in [25] may be stated as follows: If a class of multivalued maps has arbitrarily close chain approximations, then there is a fixed point index with all properties for this class.…”

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confidence: 99%

Borsuk-Ulam theorem, fixed point index and chain approximations for maps with multiplicity

Haeseler¹,

Skordev²

1992

Pacific J. Math.

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In this article we consider m-acyclic maps with respect to a field F and prove the existence of chain approximation for such maps. Furthermore we provide a unified approach to the Borsuk-Ulam theorem and the Bourgin-Yang generalization. Finally we prove the existence of ^-systems for certain m-acyclic maps and define a fixed point index.There are essentially two ways to handle multivalued fixed point or coincidence problems. The first one is based on homological arguments (homological method) and the second on the homotopical method where the multivalued map is approximated with single valued maps. For a survey of both methods we recommend [3], for single valued maps we refer to [4,8].The homological method also splits in two directions-the considerations on the level of homology groups and chain approximation techniques. For the first one see [3], the second one-chain approximations of multivalued maps-has roots in the early work of L. Vietoris (see [1] where the Vietoris-Begle mapping theorem is proved). The chain approximation technique is used by S. Eilenberg and D. Montgomery [10] to prove a Lefschetz fixed point theorem for acyclic maps on compact ANR's. B. O'Neil constructed chain approximations for a more general mapping class, i.e. (1, ?z)-mappings, and also proved a Lefschetz fixed point theorem for such mappings on polyhedra [22]. The same technique was the main tool for developing the fixed point index with all properties (including commutativity and mod-/7-ρroperty multiplicity is proved in [26]) for multivalued maps of ANR's ([9,11,25]). The main result in [25] may be stated as follows: If a class of multivalued maps has arbitrarily close chain approximations, then there is a fixed point index with all properties for this class.In this paper we consider so called m-acyclic maps with respect to a given field F and prove that for such mappings there exist chain We also provide a unified approach to Borsuk-Ulam theorems for single valued mappings as well as for m-acyclic maps with respect to ΊJ2 via chain approximations. The Borsuk-Ulam theorem or antipodal theorem can be stated in several ways; we prefer the following formulation: If /: S n -±W is a continuous map, then there exists a point

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Fixed point index and chain approximations

Cited by 31 publications

References 23 publications

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

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

Fixed Points of Arc-Component-Preserving Maps

Borsuk-Ulam theorem, fixed point index and chain approximations for maps with multiplicity

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