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.

中文翻译：

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具有自由幂零基本群的 nilmanifolds 上的微分同胚不动点

令 $M$ 是一个具有基本群的 nilmanifold，该群在至少 4 个生成器上是免费的 我们将证明对于任何非负整数 $n$ 都存在 $M$ 的自微分同胚 $h_n$ 使得 $h_n$ 正好有 $n$ 不动点和 $M$ 的任何自映射 $f$ 是$h_n$ 的同伦至少有 $n$ 固定点。我们还将阐明较少生成器和较高幂零类的情况。