Maps of tori and Pontryagin duality (Q2718001)

\textit{R. B. S. Brooks, R. F. Brown, J. Pak} and \textit{D. H. Taylor} [Proc. Am. Math. Soc. 52, 398-400 (1975; Zbl 0309.55005)] proved that for a selfmap \(f\) of the \(r\)-torus the Nielsen number equals the absolute value of the Lefschetz number. The present author gives a simple proof of this result by exploiting the fact that the \(r\)-torus \(T^{r}\) may be viewed as the dual group of \(\mathbb{Z}^{r}\) and that \(f\) is homotopic to the dual \(\widehat{\varphi}\) of \(\varphi:={f_*}_1:H_1T^{r}\to H_1T^{r}\). If one assumes that all iterates of \(f\) have non-vanishing Lefschetz number then it is shown that \(\widehat{\varphi}\) minimizes the number of fixed points in the homotopy class of \(f\).