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

Volume 235 / 2016

Evelyn L. Hart, Philip R. Heath, Edward C. Keppelmann Fundamenta Mathematicae 235 (2016), 101-126 MSC: Primary 55M20. DOI: 10.4064/fm45-1-2016 Published online: 18 July 2016

Abstract

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.

Authors

  • Evelyn L. HartDepartment of Mathematics
    Colgate University
    13 Oak Drive
    Hamilton, NY 13346-1398, U.S.A.
    e-mail
  • Philip R. HeathDepartment of Mathematics and Statistics
    Memorial University of Newfoundland
    St. John’s, NL, Canada, A1C 5S7
    e-mail
  • Edward C. KeppelmannDepartment of Mathematics
    University of Nevada – Reno
    1664 N. Virginia Street
    Reno, NV 89557-0084, U.S.A.
    e-mail

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