Linear foliations ofT n

Cohomologically rigid $\mathbb{Z}^{p}$ -actions on low-dimension tori

“…The above theorem was proved in [1] in the case where ϕ acts by translations and of course the action A is trivial. For Z p -actions by translations on T q if the first cohomology is finite dimensional then necessarily H j (Z p , C ∞ ϕ (T q )) H j (T p , Z) for 0 ≤ j ≤ p. This is not true for a general affine Z p action [8].…”

“…This gives a Z p -module structure on C ∞ (M), denoted by C ∞ ϕ (M). For the minimal Diophantine Z p -actions by translations on the torus T q in the sense of Arraut and dos Santos [1], the j th cohomology group is isomorphic to H j (T p , R) for all j, 1 ≤ j ≤ p. For this reason we say that a Z p -action ϕ on a closed orientable manifold M is cohomologically rigid (CR) if the j th cohomology group of ϕ is isomorphic to H j (T p , R) for all j, 1 ≤ j ≤ p. The investigation of the cohomology of ergodic actions of higher rank groups has attracted considerable interest in recent years due to the discovery of a number of somewhat unexpected rigidity phenomena.…”

“…A locally-free R p -action ψ is parameter rigid iff any other R p -action with the same orbit foliation is conjugate to ψ up to a reparametrization by an automorphism of R p . The only known examples of CR Z p -actions are the affine CR Z p -actions on tori [1,7], which we now describe. Thus the canonical action is parameter rigid iff the first cohomology of ϕ is isomorphic to R p .…”

On the Cohomology of Foliated Bundles

“…If α is Liouville then H is not closed and the group of co-invariants is infinite dimensional. A proof of this facts uses standard argument on Fourier series, [2].…”

“…In this article we study the cohomology of foliated bundles F → (M, [14], [15] show that the cohomology H * (F) of a foliated bundle suspension of an action ϕ : Γ → Dif f (F ) is infinite dimensional for a large class of actions. In section 4 we discuss the case B = T p and show that the cohomology of groups gives a procedure for the computation of H * (Z p , C ∞ (F )) which can be used to give an alternative simple way for computing the cohomology of linear foliations of T n , [2]. We also state R.U.…”