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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

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