Coincidence Wecken homotopies versus Wecken homotopies relative to a fixed homotopy in one of the maps

We study the 1-parameter Wecken problem versus the restricted Wecken problem, for coincidence free pairs of maps between surfaces. For this we use properties of the function space between two surfaces and of the pure braid group on two strings of a surface. When the target surface is either the 2-sphere or the torus it is known that the two problems are the same. We classify most pairs of homotopy classes of maps according to the answer of the two problems are either the same or different when the target is either projective space or the Klein bottle. Some partial results are given for surfaces of negative Euler characteristic.     

Coincidence Wecken Property for Nilmanifolds

Let f, g: X → Y be maps from a compact infra-nilmanifold X to a compact nilmanifold Y with dim X ≥ dim Y. In this note, we show that a certain Wecken type property holds, i.e., if the Nielsen number N(f, g) vanishes then f and g are deformable to be coincidence free. We also show that if X is a connected finite complex X and the Reidemeister coincidence number R(f, g) = ∞ then f ～ f' so that C(f', g) = {x ∈ X | f'(x) = g(x)} is empty.     

Equations in free groups and coincidence of mappings on surfaces

Let be a continuous map and a constant map between closed orientable surfaces of genus h,g, respectively. By definition the pair (f,c) has the Wecken property if f can be deformed into a map such that every coincidence classes of (f',c) is essential and consists of exactly one point. The main result is that (f,c) has the Wecken property if and only if where . Certain quadratic equations in free groups closely related to the coincidence problem are solved. Received January 26, 1999; in final form December 10, 1999 / Published online March 12, 2001     

Roots of mappings on nonorientable surfaces¶and equations in free groups

Let f:M ₁→M ₂ be a continuous map and c:M ₁→M ₂ a constant map between closed (not necessarily orientable) surfaces. By definition the pair (f,c) has the Wecken property if f can be deformed into a map f' such that the number of coincidence points of (f',c) is the same as the number of essential coincidence classes of (f,c) and, hence, every essential coincidence class consists of exactly one point. When both surfaces are orientable the problem to determine all maps which have the Wecken property was solved in [14]. Let A(f) denote the absolute degree as defined in [6] or [15] and . Here we show that a map f has the Wecken property iff either the Euler characteristic or . In free groups there are solved certain quadratic equations closely related to the root problem. Received: Received: 18 January 2001 / Revised version: 27 November 2001     

Preservation of the existence of fixed and coincidence points under homotopy of mappings of ordered sets

In problems of topology and analysis, well-known theorem on the preservation by any continuous homotopy of the property of a mapping to have a fixed point and the property of a pair of mappings to have a coincidence point are extensively applied. Thus, for contraction mappings and some of their generalizations, Frigon’s results on the preservation of the property to have a fixed point by a homotopy of a special type are known. This paper presents theorems on the preservation by order homotopy of the property of a pair of mappings to have a coincidence point. As a corollary, conditions under which such a homotopy preserves the property of a mapping to have a fixed point are obtained.     

BOUNDS ON COINCIDENCE INDICES ON NON-ORIENTABLE SURFACES

§ 1 . IntroductionQuestions about bounds for indices ?rst appeared in the ?xed point context. The ?rstresults appeared in studies of surface homeomorphisms (see [13, 18, 19]). In [12, 14] and[15] some results about bounds for Nielsen ?xed point class ind…     

Fixed points of <Emphasis Type="Italic">n</Emphasis>-valued maps on surfaces and the Wecken property—a configuration space approach

In this paper, we explore the fixed point theory of n-valued maps using configuration spaces and braid groups, focusing on two fundamental problems, the Wecken property, and the computation of the Nielsen number. We show that the projective plane (resp. the 2-sphere S²) has the Wecken property for n-valued maps for all n ∈ ? (resp. all n ≥ 3). In the case n = 2 and S², we prove a partial result about the Wecken property. We then describe the Nielsen number of a non-split n-valued map Open image in new window of an orientable, compact manifold without boundary in terms of the Nielsen coincidence numbers of a certain finite covering q: X? → X with a subset of the coordinate maps of a lift of the n-valued split map Open image in new window .     

Stable Homotopy Monomorphisms

It is well known that the concept of monomorphism in a category can be defined using an appropriate pullback diagram. In the homotopy category of TOP pullbacks do not generally exist. This motivated Michael Mather to introduce another notion of homotopy pullback which does exist. The aim of this paper is to investigate the modified notion of homotopy monomorphism obtained by applying the pullback characterization using Mather's homotopy pullback. The main result of Section 1 shows that these modified homotopy monomorphisms are exactly those homotopy monomorphisms (in the usual sense) which are homotopy pullback stable, hence the terminology “stable” homotopy monomorphism. We also link these stable homotopy monomorphisms to monomorphisms and products in the track homotopy category over a fixed space. In Section 2 we answer the question: when is a (weak) fibration also a stable homotopy monomorphism? In the final section it is shown that the class of (weak) fibrations with this additional property coincides with the class of “double” (weak) fibrations. The double (weak) covering homotopy property being introduced here is a stronger version of the (W) CHP in which the final maps of the homotopies involved play the same role as the initial maps.     

Multifibrations. A Class of Shape Fibrations with the Path Lifting Property

In this paper we introduce a class of maps possessing a multivalued homotopy lifting property with respect to every topological space. We call these maps multifibrations and they represent a formally stronger concept than that of shape fibration. Multifibrations have the interesting property of being characterized in a completely intrinsic way by a path lifting property involving only the total and the base space of the fibration. We also show that multifibrations (and also, with some restrictions, shape fibrations) have a lifting property for homotopies of fine multivalued maps. This implies, when the spaces considered are metric compacta, that the possibility of lifting a fine multivalued map is a property of the corresponding strong shape morphism and not of the particular map considered.     

Über einige formale Eigenschaften von Faserungen und h-Faserungen

The aim of this paper is to study some formal properties of fibrations (=continuous maps which have the covering homotopy property), cofibrations (=maps which have the homotopy extension property), h-fibrations ([2],6.4) and h-cofibrations([2],2.2). We introduce the notion of a generalized homotopy system in a category (2). This notion will be selfdual. To show that fibrations and cofibrations, h-fibrations and h-cofibrations are dual notions, no use will be made of adjoint functors. Our approach admits the transition from a categoryC to other categories, e.g. to the categoryC _B of objects over a given object B ofC.     

Homotopical theory of periodic points of periodic homeomorphisms on closed surfaces

In this work we show that the Wecken theorem for periodic points holds for periodic homeomorphisms on closed surfaces, which therefore completes the periodic point theory in such a special case. Using it we derive the set of homotopy minimal periods for such homeomorphisms. Moreover we show that the results hold for homotopically periodic self-maps of closed surfaces. This let us to re-formulate our results as a statement on properties of elements of finite order in the group of outer automorphisms of the fundamental group of a surface with non-positive Euler characteristic.     

PL fibrations are fibrations in the PL category

Proper PL maps which are Hurewicz fibrations have the covering homotopy property in the PL category.     

Absolute extensors and absolute neighborhood extensors in asymptotic categories

This paper deals with extending maps in asymptotic categories, i.e., in categories consisting of metric spaces and asymptotically Lipschitz coarsely proper maps. We demonstrate certain examples of absolute extensors and absolute neighborhood extensors. We give some conditions under which a version of Borsuk's homotopy extension theorem holds in these categories, and in answer to a problem posed by Dranishnikov in [Russian Math. Surveys 55 (2000) 1085] we show the failure of a general homotopy extension theorem. Finally, we show that a pair of an Hadamard space and its convex subspace has the homotopy extension property.     

Regularity results for real analytic homotopies

Summary In this paper, we study two main features of the homotopy curves which we follow when we use the homotopy method for solving the zeros of analytic maps. First, we prove that near the solution the curve behaves nicely. Secondly, we prove that the set of starting points which give smooth homotopy curves is open and dense. The second property is of particular importance in computer implementation.This research was supported by the National Science Foundation under Grants MCS 78-02420 (Li) and MCS-7818858 (Mallet-Paret) and MCS-7818221 (Yorke), by the Army Research office under Grant DAAG-29-80-C-0040 (Li and Yorke)     

The finiteness of the Reidemeister number of morphisms between almost-crystallographic groups

We give a practical criterion to determine whether a given pair of morphisms between almost-crystallographic groups has a finite Reidemeister coincidence number. As an application, we determine all two- and three-dimensional almost-crystallographic groups that have the R _∞ property. We also show that for a pair of continuous maps between oriented infra-nilmanifolds of equal dimension, the Nielsen coincidence number equals the Reidemeister coincidence number when the latter is finite.     

具有脉冲效应和功能反应的捕食系统的周期解

研究具有时滞、Holling-类功能反应和脉冲效应的捕食-食饵系统正周期解的存在性.利用Mawhin重合度理论和同伦性质,获得周期解存在的充分条件.结果推广和改进一些最近文献结论.     

Wecken's theorem for periodic points in dimension at least 3

Boju Jiang introduced a homotopy invariant NFn(f), for a natural number n, which is a lower bound for the cardinality of periodic points, of period n, of a self-map of a compact polyhedron. In [J. Jezierski, Wecken theorem for periodic points, Topology 42 (5) (2003) 1101-1124] and [J. Jezierski, Wecken theorem for fixed and periodic points, in: Handbook of Topological Fixed Point Theory, Kluwer Academic, Dordrecht, 2005] we prove that any self-map of a compact PL-manifold (dimM?3) is homotopic to a map g satisfying #Fix(gn)=NFn(f) i.e. NFn(f) is the best such homotopy invariant. Here we give an alternative, simpler proof of these results.     

具有时变时滞和Holling-N类功能反应的脉冲捕食系统的周期解

考虑具有时变时滞、Holling-N类功能反应和脉冲效应的捕食-食饵系统的数学模型.利用Mawhin重合度理论,并结合同伦不变性质,以及Young不等式,获得该系统正周期解存在的充分条件,推广和改进(正)最近一些文献的结果.     

On relations between gradient and classical equivariant homotopy groups of spheres

We investigate relations between stable equivariant homotopy groups of spheres in classical and gradient categories. To this end, the auxiliary category of orthogonal equivariant maps, a natural enlargement of the category of gradient maps, is used. Our result allows for describing stable equivariant homotopy groups of spheres in the category of orthogonal maps in terms of classical stable equivariant groups of spheres with shifted stems. We conjecture that stable equivariant homotopy groups of spheres for orthogonal maps and for gradient maps are isomorphic.     

Fixed point theorems for condensing admissible maps in topological vector spaces

Introducing the notion of c-measure of noncompactness due to S. Hahn, we give a homotopy extension theorem and the fixed point property for pseudo-condensing admissible maps in general topological vector spaces. Moreover, these results are applied to obtain a Leray-Schauder type fixed point theorem for pseudo-condensing admissible maps in topological vector spaces which includes many known results.