Coincidence Wecken homotopies versus Wecken homotopies relative to a fixed homotopy in one of the maps
type: Item
instance of: scholarly article
title: Coincidence Wecken homotopies versus Wecken homotopies relative to a fixed homotopy in one of the maps
zbMATH Open document ID: 1194.55004
author: M. R. Kelly
author: Daciberg Lima Gonçalves
published in: Topology and its Applications
publication date: 2010-06-09T00:00:00Z
review text: Given two self-maps which achieve the minimal number of fixed points of all maps in a homotopy class, \textit{H. Schirmer} proposed in [Pac. J. Math. 108, 191--202 (1983; Zbl 0476.54040)] a question: is there a homotopy \(H\) between them such that for each \(t\in I\), \(H_t\) gives a self-map that achieves the minimal number of fixed points? The answer turned out to be negative in general from the examples given by the second author in [Topology Appl. 37, No. 3, 293--297 (1990; Zbl 0715.55003)]. The authors of this paper extend Schirmer's question to the case of coincidence points. Let \((f_1, g_1)\), \((f_2, g_2)\) be two pairs of maps which have the minimal number of coincidence points of all pairs of maps in a given homotopy class of pairs. Two questions may be asked: (1) Can one find a homotopy \((F, G)\) connecting \((f_1, g_1)\) and \((f_2, g_2)\) such that all \((F_t, G_t)\)'s have the minimal number of coincidence points; (2) Can one find a homotopy \((F, G)\) as in (1) if \(F\) is fixed. The main results of this paper concern the equivalence of these two questions for maps between closed surfaces. Some sufficient conditions are obtained. The authors also give many concrete and interesting examples showing that these two question may not be equivalent.
Mathematics Subject Classification ID: 55M20
zbMATH DE Number: 5718377
zbMATH Keywords: coincidence point
zbMATH Keywords: function space
zbMATH Keywords: braid group
zbMATH Keywords: equation on groups
reviewed by: Xuezhi Zhao
MaRDI profile type: Publication
full work available at URL: j.topol.2010.02.027
OpenAlex ID: W1993522777
cites work: On removing coincidences of two maps when only one, rather than both, of them may be deformed by a homotopy
cites work: The Degree of a Map
cites work: Generalized Normal Bundles for Locally-Flat Imbeddings
cites work: The Nielsen number on surfaces
cites work: The braid groups of the projective plane.
cites work: Q4347420
cites work: Maps between surfaces and minimal coincidence sets for homotopies
cites work: Maps into the torus and minimal coincidence sets for homotopies
cites work: Wecken type problems for self-maps of the Klein bottle
cites work: Coincidence properties for maps from the torus to the Klein bottle
cites work: Fixed points on Klein bottle fiber bundles over the circle
cites work: The fundamental group of the space of maps from a surface into the projective plane
cites work: On fibre spaces and the evaluation map
cites work: Covering transformations and universals fibrations
cites work: Spaces of Maps into Eilenberg-Maclane Spaces
cites work: The coincidence Nielsen number for maps into real projective spaces
cites work: Some examples concerning homotopies of fixed point free maps
cites work: Fixed point sets of homotopies
cites work: Q5603827
cites work: Braid Groups of Compact 2-Manifolds with Elements of Finite Order
cites work: Q3242617
cites work: Q4192669
DOI: 10.1016/J.TOPOL.2010.02.027
Recommended article: Coincidence Wecken homotopies versus Wecken homotopies relative to a fixed homotopy in one of the maps. II
Recommended article: Q4287283
Recommended article: 𝐺-coincidences for maps of homotopy spheres into CW-complexes
Recommended article: On removing coincidences of two maps when only one, rather than both, of them may be deformed by a homotopy
Recommended article: On homotopy classes of cochain maps
Recommended article: Maps into the torus and minimal coincidence sets for homotopies
Recommended article: Q3111536
Recommended article: The incidence correspondence and its associated maps in homotopy
Recommended article: On weak homotopy equivalences between mapping spaces
Recommended article: Coincidence index for fiber-preserving maps: An approach to stable cohomotopy
description: scientific article; zbMATH DE number 5718377