Reidemeister conjugacy for finitely generated free fundamental groups

Evelyn L. Hart¹

¹ Department of Mathematics Colgate University Hamilton, NY 13346-1398, U.S.A.

Fundamenta Mathematicae, Tome 199 (2008) no. 2, pp. 93-118.

Résumé

Let $X$ be a space with the homotopy type of a bouquet of $k$ circles, and let $f:X\to X$ 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 $z\in \pi _1(X)$ to the equation $z=W_y^{-1}f_{\sharp }(z)W_x.$ 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.

Détail

DOI : 10.4064/fm199-2-1

Keywords: space homotopy type bouquet circles map certain cases algebraic techniques calculate nielsen number which homotopy invariant lower bound number fixed points maps homotopic given fixed points their corresponding group elements fixed points nielsen equivalent only there solution equation sharp nielsen number number equivalence classes have nonzero fixed point index variety methods determining nielsen classes each their own restrictions map have developed wagner kim fundamental group generators kim order describe many these methods common terminology introduce definitions describe types bounds occur best directions future research become clear nomenclature illustrate concepts extend wagners ideas regarding w characteristic maps maps remnant classes maps have only partial remnant prove these classes maps wagners algorithm almost nielsen equivalences algorithm extended nielsen equivalences proof algorithm does nielsen number complex even though these classes maps restrictive classes maps mrn maps maps number possible solutions mrn maps maps addition length solution three mrn maps maps makes computer search reasonable many examples included