Some developments in Nielsen fixed point theory. (English) Zbl 1386.55005
The authors give a survey (which they modestly call “brief”) on Nielsen fixed point theory. There are no proofs but an exhaustive bibliography comprising 81 items. They start with a short historical account beginning of course with Nielsen’s work in 1921 and proceeding to Reidemeister and Wecken and the Chinese school starting with Kiang and moving on to Shi and the first author. The authors then turn to the various ramifications of Nielsen theory such as relative, parametrised, or equivariant Nielsen theory. In the next section they deal with the computation of Nielsen numbers in homogeneous spaces and with the Reidemeister number. There is a discussion of algorithms for Nielsen numbers on surfaces and graphs and a section on modern extensions of Wecken’s minimality theorem and the minimum number of fixed points under isotopy. A very important section covers periodic points, minimal sets of periods, and zeta-functions. There is a short section on braid forcing and an even shorter one with the cryptic title “formal approaches” which deals with Lück’s universal Lefschetz invariant and K. Ponto’s generalization [Fixed point theory and trace for bicategories. Paris: Société Mathématique de France (SMF) (2010; Zbl 1207.18001)] of A. Dold’s and D. Puppe’s trace [in: Geometric topology, Proc. int. Conf., Warszawa 1978, 81–102 (1980; Zbl 0473.55008)]. For everyone wanting to learn about Nielsen theory this article written by two leading experts definitely is a “must”.
Reviewer: Christian Fenske (Gießen)
MSC:
| 55M20 | Fixed points and coincidences in algebraic topology |
| 55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |
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