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

Evelyn L. Hart¹ ; Philip R. Heath² ; Edward C. Keppelmann³

¹ Department of Mathematics Colgate University 13 Oak Drive Hamilton, NY 13346-1398, U.S.A.
² Department of Mathematics and Statistics Memorial University of Newfoundland St. John’s, NL, Canada, A1C 5S7
³ Department of Mathematics University of Nevada – Reno 1664 N. Virginia Street Reno, NV 89557-0084, U.S.A.

Fundamenta Mathematicae, Tome 235 (2016) no. 2, pp. 101-126.

Résumé

In this second and final paper of the series, we sketch an algorithm for the computation of the Nielsen type number $N\varPhi _n(f)$ for periodic points of maps $f$ of remnant 2 on surfaces with boundary, and on bouquets of circles. The number $N\varPhi _n(f)$ is the second, and more complicated, of two Nielsen type periodic point numbers. In the first paper we exhibited an algorithm for the computation of $NP_n(f)$, the first of these numbers. The class of spaces and maps under consideration in this paper do not satisfy the, by now familiar, conditions that give rise to computational theorems for $N\varPhi _n(f)$. Because of this we are thrown back on the definition which requires that we find the minimum height among all sets of $n$-representatives for $f$. Our technique, which is presented with a view to clarity rather than for efficiency of computation, is to sketch an algorithm for the construction of a finite weighted graph $(\mathcal { U}(h,n), \mathcal {D})$ whose nodes are orbits, and whose edges are the “boosts” between individual orbits. The graph $\mathcal { U}(h,n)$ is universal in the sense that the set of nodes contains all minimum sets of $n$-representatives. The graph is weighted by means of data labels $ \mathcal { D}$, which we use to determine which subset of the nodes of $\mathcal { U}(h,n)$ are sets of $n$-representatives and to compute their heights. Two factors complicate the algorithmic computation of $(\mathcal { U}(h,n), \mathcal { D})$. The first is the need to include certain nodes in $\mathcal { U}(h,n)$ that represent empty orbits, which of course are not detectable by the Reidemeister trace. The second is the need to attach data $ \mathcal { D}(P^k)$ to the nodes $P^k$ of $\mathcal { U}(h,n)$. Our method requires that we modify and extend word length arguments from the first paper. In the process we solve the twisted conjugacy problem for homomorphisms with limited cancellation among the generators. This potentially important result is surprisingly simple to prove.

Détail

DOI : 10.4064/fm45-1-2016

Keywords: second final paper series sketch algorithm computation nielsen type number varphi periodic points maps remnant surfaces boundary bouquets circles number varphi second complicated nielsen type periodic point numbers first paper exhibited algorithm computation first these numbers class spaces maps under consideration paper satisfy familiar conditions rise computational theorems varphi because thrown back definition which requires minimum height among sets n representatives technique which presented view clarity rather efficiency computation sketch algorithm construction finite weighted graph mathcal mathcal whose nodes orbits whose edges boosts between individual orbits graph mathcal universal sense set nodes contains minimum sets n representatives graph weighted means labels mathcal which determine which subset nodes mathcal sets n representatives compute their heights factors complicate algorithmic computation mathcal mathcal first include certain nodes mathcal represent empty orbits which course detectable reidemeister trace second attach mathcal nodes mathcal method requires modify extend word length arguments first paper process solve twisted conjugacy problem homomorphisms limited cancellation among generators potentially important result surprisingly simple prove