Algorithms for Nielsen type periodic numbers of maps with remnant on surfaces with boundary and on bouquets of circles I

Evelyn L. Hart ¹   ; Philip R. Heath ²   ; Edward C. Keppelmann ³

¹ Department of Mathematics Colgate University 13 Oak Drive Hamilton, NY 13346-1398, U.S.A.
² Department of Mathematics Memorial University of Newfoundland St. John's, NF, Canada A1C 5S7
³ Department of Mathematics and Statistics MS084 University of Nevada Reno, NV 89557, U.S.A.

Fundamenta Mathematicae, Tome 200 (2008) no. 2, pp. 101-132

Résumé

In this paper and its sequel we present a method that, under loose restrictions, is algorithmic for calculating the Nielsen type numbers $N{\mit\Phi} _n(f)$ and $NP_n(f)$ of self maps $f$ of hyperbolic surfaces with boundary and also of bouquets of circles. Because self maps of these surfaces have the same homotopy type as maps on wedges of circles, and the Nielsen periodic numbers are homotopy type invariant, we need concentrate only on the latter spaces. Of course the results will then automatically apply to the former spaces as well. The algorithm requires only that $f$ has minimal remnant, by which we mean that there is limited cancellation between the $f_{\ast }$ images of generators of the fundamental group. These methods often work even when the minimal remnant condition is not satisfied. Our methodology involves three separate techniques. Firstly, beginning with an endomorphism $h$ on the fundamental group, we adapt an algorithm of Wagner to our setting, allowing us to distinguish non-empty Reidemeister classes for iterates of a special representative map for $h$, which we introduce. Secondly, using techniques reminiscent of symbolic dynamics, we assign key algebraic information to the actual periodic points of this special representative. Finally, we use word length arguments to prove that the remaining information required for the calculation of $N{\mit\Phi} _n(f)$ and $NP_n(f)$ can be found with a finite computer search. We include many illustrative examples. In this first paper we give the tools we need in order to present and give the algorithm for $NP_n(f)$. All the tools introduced here will be needed in the sequel where we develop the extra tools needed in order to compute $N{\mit\Phi} _n(f)$.

Détail

DOI : 10.4064/fm200-2-1

Keywords: paper its sequel present method under loose restrictions algorithmic calculating nielsen type numbers mit phi self maps hyperbolic surfaces boundary bouquets circles because self maps these surfaces have homotopy type maps wedges circles nielsen periodic numbers homotopy type invariant concentrate only latter spaces course results automatically apply former spaces algorithm requires only has minimal remnant which mean there limited cancellation between ast images generators fundamental group these methods often work even minimal remnant condition satisfied methodology involves three separate techniques firstly beginning endomorphism fundamental group adapt algorithm wagner setting allowing distinguish non empty reidemeister classes iterates special representative map which introduce secondly using techniques reminiscent symbolic dynamics assign key algebraic information actual periodic points special representative finally word length arguments prove remaining information required calculation mit phi found finite computer search include many illustrative examples first paper tools order present algorithm tools introduced here needed sequel where develop extra tools needed order compute mit phi