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Computing Nielsen numbers. (English) Zbl 0806.55003

McCord, Christopher K. (ed.), Nielsen theory and dynamical systems. AMS- IMS-SIAM summer research conference, June 20-26, 1992, South Hadley, MA (USA), supported by the National Science Foundation. Providence, RI: American Mathematical Society. Contemp. Math. 152, 249-267 (1993).
If a map \(f:X\to X\) of a finite polyhedron has nonzero Lefschetz number \(L(f)\), then it has at least one fixed point, as does any map homotopic to it. If \(X\) has no local cutpoints and is not a surface of negative Euler characteristic, then the Nielsen number \(N(f)\) equals the minimum number of fixed points among all maps homotopic to \(f\). But, whereas by its definition \(L(f)\) can be calculated from the homomorphism \(f\) induced on the rational homology of \(X\), the computation of the more precise \(N(f)\) is much more difficult. The author considers \(N(f)\) to be computable if an algorithm exists that requires only rational homology and fundamental group information. In this survey paper, he discusses conditions on the polyhedron \(X\) such that \(N(f)\) is computable in this sense for all maps \(f:X\to X\). He presents an algorithm for the case that \(X\) has finite fundamental group and he reviews Jiang’s method for computing \(N(f)\) when \(X\) is a Jiang space, for instance a Lie group. Most of the paper is devoted to polyhedra that are homogeneous spaces of the type called infrasolvmanifolds. After a review of topological properties of such spaces, he presents algorithms for computing \(N(f)\), first in the more special cases where \(X\) is a nilmanifold or a solvmanifold, then for maps of infrasolvmanifolds in general. The paper concludes with suggestions for promising directions of further research in the computation of Nielsen numbers.
For the entire collection see [Zbl 0780.00035].

MSC:

55M20 Fixed points and coincidences in algebraic topology
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