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

TL;DR

This paper constructs specific diffeomorphisms on certain nilmanifolds with free 2-step nilpotent fundamental groups, controlling fixed points and providing insights into fixed point theory for these manifolds.

Contribution

It demonstrates the existence of diffeomorphisms with exactly n fixed points on nilmanifolds with free 2-step nilpotent fundamental groups, and relates fixed points of homotopic maps.

Findings

Existence of diffeomorphisms with exactly n fixed points

Homotopy invariance of fixed point count for these diffeomorphisms

Insights into fixed point behavior for manifolds with fewer generators or higher nilpotency

Abstract

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.