An algorithm for calculating the Nielsen number on surfaces with boundary
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- by Joyce Wagner PDF
- Trans. Amer. Math. Soc. 351 (1999), 41-62 Request permission
Abstract:
Let $f:M\to M$ be a self-map of a hyperbolic surface with boundary. The Nielsen number, $N(f)$, depends only on the induced map $f_{\#}$ of the fundamental group, which can be viewed as a free group on $n$ generators, $a_1,\dotsc ,a_n$. We determine conditions for fixed points to be in the same fixed point class and if these conditions are enough to determine the fixed point classes, we say that $f_{\#}$ is $W$-characteristic. We define an algebraic condition on the $f_{\#}(a_i)$ and show that “most” maps satisfy this condition and that all maps which satisfy this condition are $W$-characteristic. If $f_{\#}$ is $W$-characteristic, we present an algorithm for calculating $N(f)$ and prove that the inequality $|L(f)-\chi (M)|\le N(f)-\chi (M)$ holds, where $L(f)$ denotes the Lefschetz number of $f$ and $\chi (M)$ the Euler characteristic of $M$, thus answering in part a question of Jiang and Guo.References
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Additional Information
- Joyce Wagner
- Affiliation: Department of Mathematics, California State University, Long Beach, California 90840
- Email: pslavich@aol.com
- Received by editor(s): December 15, 1995
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 41-62
- MSC (1991): Primary 55M20
- DOI: https://doi.org/10.1090/S0002-9947-99-01827-9
- MathSciNet review: 1401531