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Let $M$ be a nilmanifold with a fundamental group which is free $2$-stepnilpotent on at least 4 generators. We will show that for any nonnegativeinteger $n$ there exists a self-diffeomorphism $h_n$ of $M$ such that $h_n$ hasexactly $n$ 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 thesituation for less generators and also for higher nilpotency classes.

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