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Nielsen numbers of maps of aspherical figure-eight type polyhedra

Nielsen numbers of maps of aspherical figure-eight type polyhedra Abstract Let X be an aspherical polyhedron of the homotopy type of the figure-eight and let f : X → X 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 π 1 ( X ) and a , b are generators of π 1 ( X ). Let f # ( b ) = U and f # ( a ) = U n R , 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 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Nielsen numbers of maps of aspherical figure-eight type polyhedra

Yi, Peter; Kim, Seung Won
Forum Mathematicum , Volume 27 (3) – May 1, 2015
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Publisher
de Gruyter
Copyright
Copyright © 2015 by the
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2012-0036
Publisher site
See Article on Publisher Site

Abstract

Abstract Let X be an aspherical polyhedron of the homotopy type of the figure-eight and let f : X → X 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 π 1 ( X ) and a , b are generators of π 1 ( X ). Let f # ( b ) = U and f # ( a ) = U n R , 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 .

Journal

Forum Mathematicum – de Gruyter

Published: May 1, 2015

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