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Reidemeister conjugacy for finitely generated free fundamental groups

100%

Hart E.

Fundamenta Mathematicae

2008

tom 199

nr 2

93-118

Let X be a space with the homotopy type of a bouquet of k circles, and let f: X → X be a map. In certain cases, algebraic techniques can be used to calculate N(f), the Nielsen number of f, which is a homotopy invariant lower bound on the number of fixed points for maps homotopic to f. Given two fixed points of f, x and y, and their corresponding group elements, $W_x$ and $W_y$, the fixed points are Nielsen equivalent if and only if there is a solution z ∈ π₁(X) to the equation $z = W_{y}^{-1}f_{♯}(z)W_{x}$. The Nielsen number is the number of equivalence classes that have nonzero fixed point index. A variety of methods for determining the Nielsen classes, each with their own restrictions on the map f, have been developed by Wagner, Kim, and (when the fundamental group is free on two generators) by Kim and Yi. In order to describe many of these methods with a common terminology, we introduce new definitions that describe the types of bounds on |z| that can occur. The best directions for future research become clear when this new nomenclature is used. To illustrate the new concepts, we extend Wagner's ideas, regarding W-characteristic maps and maps with remnant, to two new classes of maps that have only partial remnant. We prove that for these classes of maps Wagner's algorithm will find almost all Nielsen equivalences, and the algorithm is extended to find all Nielsen equivalences. The proof that our algorithm does find the Nielsen number is complex even though these two classes of maps are restrictive. For our classes of maps (MRN maps and 2C3 maps), the number of possible solutions z is at most 11 for MRN maps and 14 for 2C3 maps. In addition, the length of any solution is at most three for MRN maps and four for 2C3 maps. This makes a computer search reasonable. Many examples are included.

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

51%

Hart E., Heath P., Keppelmann E.

Fundamenta Mathematicae

2008

tom 200

nr 2

101-132

In this paper and its sequel we present a method that, under loose restrictions, is algorithmic for calculating the Nielsen type numbers NΦₙ(f) and NPₙ(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⁎ 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 NΦₙ(f) and NPₙ(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 NPₙ(f). All the tools introduced here will be needed in the sequel where we develop the extra tools needed in order to compute NΦₙ(f).

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Hart E.

1 Heath P.

1 Keppelmann E.

2 2008

Wyniki wyszukiwania

Reidemeister conjugacy for finitely generated free fundamental groups

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