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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 27, Issue 3


Nielsen numbers of maps of aspherical figure-eight type polyhedra

Peter Yi / Seung Won Kim
Published Online: 2013-02-20 | DOI: https://doi.org/10.1515/forum-2012-0036


Let X be an aspherical polyhedron of the homotopy type of the figure-eight and let f : XX be a self-map. The Wagner algorithm [Trans. Amer. Math. Soc. 351 (1999), 41–62] provides computations for the Nielsen number of self-maps of X satisfying the remnant condition. If f is without remnant, then using the concept of mutant by Jiang [Math. Ann. 311 (1998), 467–479] we may assume that f#(b) is an initial segment of f#(a), where f# is the induced endomorphism of π1(X) and a, b are generators of π1(X). Let f#(b) = U and f#(a) = UnR, where n is the maximal such positive integer. If R is not an initial segment of U, we say that f is of Type Y. In this paper, we prove that if f is of Type Y, then f can be mutated either to a map that has remnant or to an exceptional form for which we can calculate the Nielsen number directly. Not all self-maps of X are of Type Y. However, making use of the results in this paper, an algorithm is presented by Kim [J. Pure Appl. Algebra 216 (2012), 1652–1666] that does compute the Nielsen number for all self-maps of X.

Keywords: Fixed point; Nielsen number; mutant; standard form; remnant

MSC: 55M20; 20F10

About the article

Received: 2011-09-26

Revised: 2013-01-06

Published Online: 2013-02-20

Published in Print: 2015-05-01

Funding Source: National Research Foundation of Korea

Award identifier / Grant number: 2009-0077164

Citation Information: Forum Mathematicum, Volume 27, Issue 3, Pages 1277–1307, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2012-0036.

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