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