Computation of Nielsen numbers for maps of closed surfaces
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- by O. Davey, E. Hart and K. Trapp PDF
- Trans. Amer. Math. Soc. 348 (1996), 3245-3266 Request permission
Abstract:
Let $X$ be a closed surface, and let $f: X \rightarrow X$ be a map. We would like to determine $\text {Min}(f):= \mathrm {min} \{ | \mathrm {Fix}g| : g \simeq f\}.$ Nielsen fixed point theory provides a lower bound $N(f)$ for $\text {Min}(f)$, called the Nielsen number, which is easy to define geometrically and is difficult to compute. We improve upon an algebraic method of calculating $N(f)$ developed by Fadell and Husseini, so that the method becomes algorithmic for orientable closed surfaces up to the distinguishing of Reidemeister orbits. Our improvement makes tractable calculations of Nielsen numbers for many maps on surfaces of negative Euler characteristic. We apply the improved method to self-maps on the connected sum of two tori including classes of examples for which no other method is known. We also include the application of this algebraic method to maps on the Klein bottle $K$. Nielsen numbers for maps on $K$ were first calculated (geometrically) by Halpern. We include a sketch of Halpern’s never published proof that $N(f)= \text {Min}(f)$ for all maps $f$ on $K$.References
- Bestvina, M. and Handel, M. Train tracks and surface homeomorphisms, Topology 34 (1995), 109–140.
- Bosma, W., Cannon, J.$\,$J., and Mathews, G. Programming with algebraic structures: Design of the Magma language, In: M. Giesbrecht (ed), Proceedings of the 1994 International Symposium on Symbolic and Algebraic Computation, Oxford, July 20–22, 1994. Association for Computing Machinery, 1994, 52–57.
- Robin B. S. Brooks, Robert F. Brown, Jingyal Pak, and Douglas H. Taylor, Nielsen numbers of maps of tori, Proc. Amer. Math. Soc. 52 (1975), 398–400. MR 375287, DOI 10.1090/S0002-9939-1975-0375287-X
- Brouwer, L. Über die Minimalzahl der Fixpunkte bei den Klassen von eindeutigen stetigen Transformationen der Ringflächen, Math. Ann. 82 (1921), 94–96.
- Robert F. Brown, The Lefschetz fixed point theorem, Scott, Foresman & Co., Glenview, Ill.-London, 1971. MR 0283793
- Robert F. Brown, Wecken properties for manifolds, Nielsen theory and dynamical systems (South Hadley, MA, 1992) Contemp. Math., vol. 152, Amer. Math. Soc., Providence, RI, 1993, pp. 9–21. MR 1243467, DOI 10.1090/conm/152/01315
- Richard H. Crowell and Ralph H. Fox, Introduction to knot theory, Ginn and Company, Boston, Mass., 1963. Based upon lectures given at Haverford College under the Philips Lecture Program. MR 0146828
- Edward Fadell and Sufian Husseini, The Nielsen number on surfaces, Topological methods in nonlinear functional analysis (Toronto, Ont., 1982) Contemp. Math., vol. 21, Amer. Math. Soc., Providence, RI, 1983, pp. 59–98. MR 729505, DOI 10.1090/conm/021/729505
- John Franks and MichałMisiurewicz, Cycles for disk homeomorphisms and thick trees, Nielsen theory and dynamical systems (South Hadley, MA, 1992) Contemp. Math., vol. 152, Amer. Math. Soc., Providence, RI, 1993, pp. 69–139. MR 1243471, DOI 10.1090/conm/152/01319
- Halpern, B. Periodic points on the Klein bottle, preprint, 1978.
- Hart, E. An algebraic study of local Nielsen fixed point theory, Doctoral Dissertation, University of Wisconsin-Madison, 1991.
- Heath, P., Keppelmann, E., and Wong, P. Addition formulae for Nielsen numbers and for Nielsen type numbers of fibre preserving maps, Topology Appl. 67 (1995), 133–157.
- Hopf, H. Über Mindestzahlen von Fixpunkten, Math. Z., 26, 1927, 762–774.
- S. Y. Husseini, Generalized Lefschetz numbers, Trans. Amer. Math. Soc. 272 (1982), no. 1, 247–274. MR 656489, DOI 10.1090/S0002-9947-1982-0656489-X
- Bo Ju Jiang, Lectures on Nielsen fixed point theory, Contemporary Mathematics, vol. 14, American Mathematical Society, Providence, R.I., 1983. MR 685755, DOI 10.1090/conm/014
- Bo Ju Jiang, Commutativity and Wecken properties for fixed points on surfaces and $3$-manifolds, Topology Appl. 53 (1993), no. 2, 221–228. MR 1247678, DOI 10.1016/0166-8641(93)90138-4
- Michael R. Kelly, The relative Nielsen number and boundary-preserving surface maps, Pacific J. Math. 161 (1993), no. 1, 139–153. MR 1237142, DOI 10.2140/pjm.1993.161.139
- Kelly, M. Computing Nielsen numbers of surface homeomorphisms, Topology, to appear.
- Kiang Tsai-han, The theory of fixed point classes, Translated from the second Chinese edition, Springer-Verlag, Berlin; Science Press Beijing, Beijing, 1989. MR 1002187, DOI 10.1007/978-3-642-68133-2
- Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
- Christopher McCord, Nielsen numbers and Lefschetz numbers on solvmanifolds, Pacific J. Math. 147 (1991), no. 1, 153–164. MR 1081679, DOI 10.2140/pjm.1991.147.153
- Christopher K. McCord, Estimating Nielsen numbers on infrasolvmanifolds, Pacific J. Math. 154 (1992), no. 2, 345–368. MR 1159516, DOI 10.2140/pjm.1992.154.345
- Nielsen, J. Über die Minimalzahl der Fixpunkte bei Abbildungstypen der Ringflächen, Math. Ann. 82 (1921), 83–93.
- Nielsen, J. Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, I, Acta Math. 50 (1927), 189–358.
- Wagner, J. An algorithm for calculating the Nielsen number on surfaces with boundary, preprint.
Additional Information
- O. Davey
- Affiliation: Department of Mathematics, Binghamton University, Binghamton, New York 13902-6000
- Email: owen@math.binghamton.edu
- E. Hart
- Affiliation: Department of Mathematics, Hope College, Holland, Michigan 49423-9000
- Address at time of publication: Department of Mathematics, Colgate University, Hamilton, New York 13346-1398
- Email: ehart@colgate.edu
- K. Trapp
- Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551
- Email: trapp@dartmouth.edu
- Received by editor(s): October 3, 1995
- Additional Notes: The authors were partially supported by NSF grant #DMS9322328.
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 3245-3266
- MSC (1991): Primary 55M20, 57M20; Secondary 57M05
- DOI: https://doi.org/10.1090/S0002-9947-96-01693-5
- MathSciNet review: 1370638