Electronic Version 19 (1994), 149-160
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Electronic Version 19 (1994), 149-160 | |
M.R. Kelly
Given a compact polyhedron X and a self-mapping f:X --> X the Nielsen number N(f) provides a lower bound for the number of fixed points of f. This number is a homotopy invariant and in a variety of settings can be realized geometrically. That is, there exists a mapping homotopic to f which has exactly N(f) fixed points. For example, this occurs when X is a topological manifold and f is a homeomorphism [JG].
As the Nielsen number is a topological invariant, it would be nice to have computational tools at hand which would allow one to compute its value. But in general this seems to be a difficult task, even in very restricted settings such as for self-homeomorphisms of manifolds. In the paper [M] a number of instances where the Nielsen number can be effectively computed by an algorithmic procedure are presented. One such is the case when the fundamental group \pi1(X) is finite. In [K] an algorithm for the computation of N(f) is given when X is a compact surface and f is a homeomorphism. An important step in the implementation of this algorithm is an algorithm due to Bestvina and Handel [BH].
The purpose of this paper is to describe an algorithmic procedure for computing the value of N(f) when the space X is taken from a certain class of 3-manifolds and f is a homeomorphism. The first algorithm applies to Seifert fibre spaces. Here it is shown how the computation reduces to computing the Nielsen number for the induced map on the underlying orbifold. Since this orbifold is a surface, the algorithm in [K] can be applied. The second algorithm is mainly an application of a result of Jiang and Wang [JW] which gives a characterization of fixed point classes for self-homeomorphisms of manifolds taken from a class of aspherical geometric 3-manifolds.
volume 19: table of contents
topology proceedings
Electronic Version