 Self-coincidence number of a map of nonzero degree

For $f:M\to N$ between closed connected oriented manifolds of equal dimension, cyclic invariance of trace and $f^{!}{f}^{\ast }=(\mathrm{deg}f)\mathrm{id}$ give the formula. Thus if both the degree and the target Euler characteristic are nonzero, $f$ coincides somewhere with every map homotopic to it, by the Lefschetz coincidence theorem. For target ${\mathbb{CP}}^{2k}$ the Euler characteristic is $2k+1$.