This paper centers around two basic problems of topological coincidence theory. First, try to measure (with the help of Nielsen and minimum numbers) how far a given pair of maps is from being loose, i.e. from being homotopic to a pair of coincidence free maps. Secondly, describe the set of loose pairs of homotopy classes. We give a brief (and necessarily very incomplete) survey of some old and new advances concerning the first problem. Then we attack the second problem mainly in the setting of homotopy groups. This leads also to a very natural filtration of all homotopy sets. Explicit calculations are carried out for maps into spheres and projective spaces.
@article{108022,
author = {Ulrich Koschorke},
title = {Coincidence free pairs of maps},
journal = {Archivum Mathematicum},
volume = {042},
year = {2006},
pages = {105-117},
zbl = {1164.55300},
mrnumber = {2322402},
language = {en},
url = {http://dml.mathdoc.fr/item/108022}
}
Koschorke, Ulrich. Coincidence free pairs of maps. Archivum Mathematicum, Tome 042 (2006) pp. 105-117. http://gdmltest.u-ga.fr/item/108022/
Wecken properties for manifolds, Contemp. Math. 152 (1993), 9–21. (1993) | MR 1243467 | Zbl 0816.55001
Nielsen coincidence theory and coincidence–producing maps for manifolds with boundary, Topology Appl. 46 (1992), 65–79. (1992) | MR 1177164 | Zbl 0757.55002
On removing coincidences of two maps when only one, rather than both, of them may be deformed by a homotopy, Pacif. J. Math. 39 (1971), 45–52. (1971) | MR 0341470
Self–coincidence of fibre maps, Osaka J. Math. 42 (2005), 291–307. | MR 2147735 | Zbl 1079.55006
Configuration spaces, Math. Scand. 10 (1962), 111–118. (1962) | MR 0141126 | Zbl 0136.44104
Nielsen Fixed Point Theory, Handbook of geometric topology, (R.J. Daverman and R. B. Sher, Ed.), 500–521, Elsevier Science 2002. | MR 1886677 | Zbl 0992.55002
Fixed points and braids, Invent. Math. 75 (1984), 69–74. (1984) | MR 0728139 | Zbl 0565.55005
Fixed points and braids. II, Math. Ann. 272 (1985), 249–256. (1985) | MR 0796251 | Zbl 0617.55001
Commutativity and Wecken properties for fixed points of surfaces and 3–manifolds, Topology Appl. 53 (1993), 221–228. (1993) | MR 1247678
The relative Nielsen number and boundary-preserving surface maps, Pacific J. Math. 161, No. 1, (1993), 139–153. (1993) | MR 1237142 | Zbl 0794.55002
Vector fields and other vector bundle monomorphisms – a singularity approach, Lect. Notes in Math. 847 (1981), Springer–Verlag. (1981) | MR 0611333
Selfcoincidences in higher codimensions, J. reine angew. Math. 576 (2004), 1–10. | MR 2099198 | Zbl 1054.57030
Nielsen coincidence theory in arbitrary codimensions, J. reine angew. Math. (to appear). | MR 2270573 | Zbl 1107.55002
Linking and coincidence invariants, Fundamenta Mathematicae 184 (2004), 187–203. | MR 2128050 | Zbl 1116.57021
Geometric and homotopy theoretic methods in Nielsen coincidence theory, Fixed Point Theory and App. (2006), (to appear). | MR 2210381 | Zbl 1094.55006
Nonstabilized Nielsen coincidence invariants and Hopf–Ganea homomorphisms, Geometry and Topology 10 (2006), 619–665. | MR 2240900 | Zbl 1128.55002
Minimizing coincidence numbers of maps into projective spaces, submitted to Geometry and Topology monographs (volume dedicated to Heiner Zieschang). (The papers [12] – [17] can be found at http://www.math.uni-siegen.de/topology/publications.html) | MR 2484710 | Zbl 1158.55002
Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, Acta Math. 50 (1927), 189–358. (1927) | MR 1555256
Composition methods in homotopy groups of spheres, Princeton Univ. Press, 1962. (1962) | MR 0143217 | Zbl 0101.40703
Fixpunktklassen, I, Math. Ann. 117 (1941), 659–671. (1941) | MR 0005339
Fixpunktklassen, II, Math. Ann. 118 (1942), 216–234. (1942) | MR 0010280
Fixpunktklassen, III, Math. Ann. 118 (1942), 544–577. (1942) | MR 0010281