The Reidemeister trace and the calculation of the Nielsen number

Abstract

1. Introduction. One goal for Nielsen fixed point theory is the capturing of geometric information in an algebraic form. The Reidemeister trace for a self-map f on a finite, connected CW complex X is an alternating sum of algebraic traces involving the cellular chains for the universal cover of X. This algebraic formal sum, when it is reduced, has for each term the index of an algebraic Nielsen class multiplied by the class itself. Thus the Nielsen number N (f ) is the number of terms with non-zero coefficient, and the Lefschetz number L(f ) is the sum of the coefficients. Wecken proved in [We, part II] that this connection between the algebraic sum and the geometric Nielsen number holds for X a compact, connected polyhedron. Modern treatments of Wecken's work extend this connection to CW complexes (see [Hu]) and to finitely dominated spaces (see [G1]). We provide a sketch of the proof for Wecken's setting and discuss the modern treatments in Section 3. We discuss the difficulties involved in calculating the Reidemeister trace and mention some results that address these problems in Section 5. We also provide in Section 6 references for papers that include definitions of Reidemeister traces for many of the variations on Nielsen fixed point theory.Reidemeister first defined the alternating sum of traces (see [R]) that Wecken named the Reidemeister trace ("die Reidemeistersche Spureninvariante," [We, part II]). Later the trace was also called the generalized Lefschetz number. At the conference that gave rise to these proceedings, it was generally agreed to return to the original name. Thanks are due to R. Geoghegan for helpful conversations.Just as the Lefschetz number is defined as an alternating sum of algebraic traces and can provide information about the existence of a fixed point, the Reidemeister trace is

Estimating Nielsen Numbers on Wedge Product Spaces

“…Then f = ψ • ϕ and from the commutativity of the Nielsen number, we have 9) where ϕ • ψ is a map from S to itself. Thus calculating the Nielsen number N( f ) for any map which satisfies condition (H2) is now reduced to the calculation of the Nielsen number on surfaces with boundary.…”

“…In order to compute the Nielsen number of f , we use the Fadell-Husseini formula for the Reidemeister trace in [8]. (See [8] or [9].) For the map f , the Reidemeister trace is …”

Reidemeister conjugacy for finitely generated free fundamental groups

“…A detailed description of the necessary background can be found in [5] and [6], and the reader is encouraged to refer to these sources for details that are omitted here.…”

“…Once the terms of the Fox trace are collected into Reidemeister orbits, the resulting element of the free Z-module generated by the Reidemeister orbits is equal to the Reidemeister trace (or the generalized Lefschetz number). See, for example, [5]. The Nielsen number is the number of terms with nonzero coefficient in the Reidemeister trace.…”

Equivariant Nielsen invariants for discrete groups

“…In contrast to the generalized equivariant Lefschetz invariant λ G (f ) these do not incorporate the W H-action on the fixed point set X H . A generalized Lefschetz trace for equivariant maps has also been defined by Wong [Won], as mentioned in [Har99].…”