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Transactions of the American Mathematical Society

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



An algorithm for calculating
the Nielsen number
on surfaces with boundary

Author: Joyce Wagner
Journal: Trans. Amer. Math. Soc. 351 (1999), 41-62
MSC (1991): Primary 55M20
MathSciNet review: 1401531
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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.

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Joyce Wagner
Affiliation: Department of Mathematics, California State University, Long Beach, California 90840

Received by editor(s): December 15, 1995
Article copyright: © Copyright 1999 American Mathematical Society