Geodesic
Coincidence free pairs of maps
Archivum mathematicum, Tome 42 (2006) no. 5, pp. 105-117
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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.
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.
Classification : 55M20, 55Q52, 57R22
Keywords: coincidence; Nielsen number; minimum number; configuration space; projective space; filtration 
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Koschorke, Ulrich. Coincidence free pairs of maps. Archivum mathematicum, Tome 42 (2006) no. 5, pp. 105-117. http://geodesic.mathdoc.fr/item/ARM_2006_42_5_a2/

[1] Brown R.: Wecken properties for manifolds. Contemp. Math. 152 (1993), 9–21. | MR | Zbl

[2] Brown R., Schirmer H.: Nielsen coincidence theory and coincidence–producing maps for manifolds with boundary. Topology Appl. 46 (1992), 65–79. | MR | Zbl

[3] Brooks R. B. S.: 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. | MR

[4] Dold A., Gonçalves D.: Self–coincidence of fibre maps. Osaka J. Math. 42 (2005), 291–307. | MR | Zbl

[5] Fadell E., Neuwirth L.: Configuration spaces. Math. Scand. 10 (1962), 111–118. | MR | Zbl

[6] Geoghegan R.: Nielsen Fixed Point Theory. Handbook of geometric topology, (R.J. Daverman and R. B. Sher, Ed.), 500–521, Elsevier Science 2002. | MR | Zbl

[7] Jiang B.: Fixed points and braids. Invent. Math. 75 (1984), 69–74. | MR | Zbl

[8] Jiang B.: Fixed points and braids. II. Math. Ann. 272 (1985), 249–256. | MR | Zbl

[9] Jiang B.: Commutativity and Wecken properties for fixed points of surfaces and 3–manifolds. Topology Appl. 53 (1993), 221–228. | MR

[10] Kelly M.: The relative Nielsen number and boundary-preserving surface maps. Pacific J. Math. 161, No. 1, (1993), 139–153. | MR | Zbl

[11] Koschorke U.: Vector fields and other vector bundle monomorphisms – a singularity approach. Lect. Notes in Math. 847 (1981), Springer–Verlag. | MR

[12] Koschorke U.: Selfcoincidences in higher codimensions. J. reine angew. Math. 576 (2004), 1–10. | MR | Zbl

[13] Koschorke U.: Nielsen coincidence theory in arbitrary codimensions. J. reine angew. Math. (to appear). | MR | Zbl

[14] Koschorke U.: Linking and coincidence invariants. Fundamenta Mathematicae 184 (2004), 187–203. | MR | Zbl

[15] Koschorke U.: Geometric and homotopy theoretic methods in Nielsen coincidence theory. Fixed Point Theory and App. (2006), (to appear). | MR | Zbl

[16] Koschorke U.: Nonstabilized Nielsen coincidence invariants and Hopf–Ganea homomorphisms. Geometry and Topology 10 (2006), 619–665. | MR | Zbl

[17] Koschorke U.: 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 | Zbl

[18] Nielsen J.: Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. Acta Math. 50 (1927), 189–358. | MR

[19] Toda H.: Composition methods in homotopy groups of spheres. Princeton Univ. Press, 1962. | MR | Zbl

[20] Wecken F.: Fixpunktklassen, I. Math. Ann. 117 (1941), 659–671. | MR

Wecken F.: Fixpunktklassen, II. Math. Ann. 118 (1942), 216–234. | MR

Wecken F.: Fixpunktklassen, III. Math. Ann. 118 (1942), 544–577. | MR