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

Abstract:

We show that for each natural $p\geq 2$, 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.