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We study cohomology-free (c.f.) diffeomorphisms of the torus $T^n$. A diffeomorphism is c.f. if every smooth function $f$ on $T^n$ is cohomologous to a constant $f_0$, i.e. there exists a $C^{\infty}$ function $h$ so that $h-h\circ\varphi=f-f_0$. We show that the only c.f. diffeomorphisms of $T^n$, $1\le n\le3$, are the smooth conjugations of Diophantine translations. For $n=4$, we prove the same result for c.f. orientation-preserving diffeomorphisms.

A locally free action of a Lie group is parameter rigid if for each other action with the same orbit foliation there exists a $C^\infty $ orbit-preserving diffeomorphism which conjugates the action to a reparametrization of the other by an automorphism of the Lie group. We show that for actions of the Heisenberg groups, if the first leafwise cohomology group of the orbit foliation is isomorphic to the first cohomology of the Lie algebra of the group, then the action is parameter rigid. Using this, we give examples of parameter rigid actions for all of the Heisenberg groups.

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