Research Article | Open | Published:
Wecken type problems for self-maps of the Klein bottle
Fixed Point Theory and Applicationsvolume 2006, Article number: 75848 (2006)
We consider various problems regarding roots and coincidence points for maps into the Klein bottle . The root problem where the target is and the domain is a compact surface with non-positive Euler characteristic is studied. Results similar to those when the target is the torus are obtained. The Wecken property for coincidences from to is established, and we also obtain the following 1-parameter result. Families which are coincidence free but any homotopy between and , , creates a coincidence with . This is done for any pair of maps such that the Nielsen coincidence number is zero. Finally, we exhibit one such family where is the constant map and if we allow for homotopies of , then we can find a coincidence free pair of homotopies.
Bogatyi S, Gonçalves DL, Zieschang H: The minimal number of roots of surface mappings and quadratic equations in free groups. Mathematische Zeitschrift 2001,236(3):419–452.
Brooks RBS: On removing coincidences of two maps when only one, rather than both, of them may be deformed by a homotopy. Pacific Journal of Mathematics 1972,40(1):45–52.
Brooks RBS, Brown RF, Schirmer H: The absolute degree and the Nielsen root number of compositions and Cartesian products of maps. Topology and Its Applications 2001,116(1):5–27. 10.1016/S0166-8641(00)00089-4
Brooks RBS, Odenthal C: Nielsen numbers for roots of maps of aspherical manifolds. Pacific Journal of Mathematics 1995,170(2):405–420.
Brouwer LEJ: Über die Minimalzahl der Fixpunkte bei den Klassen von eindeutigen stetigen Transformationen der Ringlfächen. Mathematische Annalen 1920,82(1–2):94–96. 10.1007/BF01457978
Brown RF, Schirmer H: Nielsen root theory and Hopf degree theory. Pacific Journal of Mathematics 2001,198(1):49–80. 10.2140/pjm.2001.198.49
Dimovski D, Geoghegan R: One-parameter fixed point theory. Forum Mathematicum 1990,2(2):125–154.
Dobreńko R, Jezierski J: The coincidence Nielsen number on nonorientable manifolds. Rocky Mountain Journal of Mathematics 1993,23(1):67–85. 10.1216/rmjm/1181072611
Epstein DBA: The degree of a map. Proceedings of the London Mathematical Society. Third Series 1966, 16: 369–383. 10.1112/plms/s3-16.1.369
Geoghegan R, Nicas A: Parametrized Lefschetz-Nielsen fixed point theory and Hochschild homology traces. American Journal of Mathematics 1994,116(2):397–446. 10.2307/2374935
Gonçalves DL: Coincidence of maps between surfaces. Journal of the Korean Mathematical Society 1999,36(2):243–256.
Gonçalves DL, Kelly MR: Maps between surfaces and minimal coincidence sets for homotopies. Topology and Its Applications 2001,116(1):91–102. 10.1016/S0166-8641(00)00084-5
Gonçalves DL, Kelly MR: Maps into the torus and minimal coincidence sets for homotopies. Fundamenta Mathematicae 2002,172(2):99–106. 10.4064/fm172-2-1
Jezierski J: One codimensional Wecken type theorems. Forum Mathematicum 1993,5(5):421–439.
Kelly MR: Some examples concerning homotopies of fixed point free maps. Topology and Its Applications 1990,37(3):293–297. 10.1016/0166-8641(90)90028-Z
Schirmer H: Fixed point sets of homotopies. Pacific Journal of Mathematics 1983,108(1):191–202.
Scott GP: Braid groups and the group of homeomorphisms of a surface. Proceedings of the Cambridge Philosophical Society 1970, 68: 605–617. 10.1017/S0305004100076593
Skora R: The degree of a map between surfaces. Mathematische Annalen 1987,276(3):415–423. 10.1007/BF01450838
Wong P: Coincidences of maps into homogeneous spaces. Manuscripta Mathematica 1999,98(2):243–254. 10.1007/s002290050137