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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 $z=W_y^{-1}{f}_{♯}\left(z\right)W_x$. The Nielsen number is the...

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...