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

## TL;DR

This paper demonstrates that for each natural number p ≥ 2, the Lefschetz fixed point theorem's application to free ${Z}^p$-actions on the 3-torus is optimal, and constructs such actions with prescribed homological properties.

## Contribution

It establishes the existence of free ${Z}^p$-actions on ${T}^3$ with specified homological actions, showing the optimality of Lefschetz fixed point theorem in this context.

## Key findings

- Existence of free ${Z}^p$-actions with prescribed homological action
- Optimality of Lefschetz fixed point theorem for these actions
- Normal form characterization of the actions

## Abstract

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.