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In order to compute the Nielsen number N(f) of a self-map f: X → X, some Reidemeister classes in the fundamental group ${\pi }_1\left(X\right)$ need to be distinguished. In this paper some algebraic results are given which allow distinguishing Reidemeister classes and hence computing the Reidemeister number of some maps. Examples of computations are presented.

Let A, $X_1$ and $X_2$ be topological spaces and let $i_1:A\to X_1$, $i_2:A\to X_2$ be continuous maps. For all self-maps $f_A:A\to A$, $f_1:X_1\to X_1$ and $f_2:X_2\to X_2$ such that $f_1{i}_1=i_1{f}_A$ and $f_2{i}_2=i_2{f}_A$ there exists a pushout map f defined on the pushout space $X_1{\bigsqcup }_A{X}_2$. In [F] we proved a formula relating the generalized Lefschetz numbers of f, $f_A$, $f_1$ and $f_2$. We had to assume all the spaces involved were connected because in the original definition of the generalized Lefschetz number given by Husseini in [H] the space was assumed to be connected. So, to extend the result of [F] to the not...