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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Computation of Nielsen numbers
for maps of closed surfaces


Authors: O. Davey, E. Hart and K. Trapp
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
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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 $K$.


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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

DOI: https://doi.org/10.1090/S0002-9947-96-01693-5
Keywords: Nielsen fixed point theory, Topological fixed point theory, Lefschetz number
Received by editor(s): October 3, 1995
Additional Notes: The authors were partially supported by NSF grant #DMS9322328.
Article copyright: © Copyright 1996 American Mathematical Society