Free ${\mathbb{Z}}^p$-actions on the three dimensional torus

Abstract

We show that for each natural $p\ge 2$, the Lefschetz fixed point theorem is optimal when applied to ${\mathbb{Z}}^p$-actions by homeomorphisms on the three dimensional torus ${\mathbb{T}}^3$. More precisely, we show that for a spectrally unitary ${\mathbb{Z}}^p$-action $\mathbf{A}$ on the first homology group $H_1({\mathbb{T}}^3,\mathbb{Z})$ with trivial fixed point set, there exists a free ${\mathbb{Z}}^p$-action by real analytic diffeomorphisms of ${\mathbb{T}}^3$ whose induced ${\mathbb{Z}}^p$-action on $H_1({\mathbb{T}}^3,\mathbb{Z})$ is the action $\mathbf{A}$. In particular, we establish the normal form for this type of actions.

14 pages, 0 figures