Fixed points of diffeomorphisms on nilmanifolds with a free nilpotent fundamental group

Let M be a nilmanifold with a fundamental group which is free 2-step nilpotent on at least 4 generators. We will show that for any nonnegative integer n there exists a self-diffeomorphism h_n of M such that h_n has exactly n fixed points and any self-map f of M which is homotopic to h_n has at least n fixed points. We will also shed some light on the situation for less generators and also for higher nilpotency classes.status: publishe