Abstract
We present a definition of diophantine matrix and use this concept to distinguish two classes of minimal linear foliations ofT n, the diophantine and the Liouville one. Let ε p , 1≤p≤n−1, denote a minimal (all leaves are dense) linearp-dimensional foliation ofT n, andH om(T n, ε p ), 1≤m≤p, the cohomology group of type (0,m) of the foliated manifold (T n, ε p ). Our main result is the computation of these groups.H om(T n, ε p ) is isomorphic to\(\mathbb{R}^{(\begin{array}{*{20}c} n \\ m \\ \end{array} )} \) if ε p is diophantine and is an infinite dimensional non-Hausdorff vector space if ε p is Liouville. Some of these groups were computed before, see [4], [6] and [9].
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References
J. L. Arraut, N. M. dos Santos, Actions of ℝp on closed manifolds, Topology Appl.29 (1988), 41–54.
J. L. Arraut, N. M. dos Santos, Differentiable Conjugation of Actions of ℝp, Bol. Soc. Bras. Mat.19 (1988), 1–19.
J. L. Arraut, N. M. dos Santos,Actions of Cylinders, Topology Appl.36 (1990), 57–71.
Aziz El Kacimi-Alaoui, A. Tihami,Cohomologie Bigraduée de Certains Feuilletages, Bull. Soc. Math. Belge, sérieB-38 (1986), 144–156.
P. Erdös,Representations of real numbers as sums and products of Liouville numbers, Mich. Math. J.9 (1962), 59–60.
J. L. Heitsch,A Cohomology for Foliated Manifolds, Comment. Math. Helvetici50 (1975), 197–218.
M. R. Herman,Sur la Conjugaison Différentiable des Diffeomorphismes du Cercle a des Rotations, Publications Mathematiques49. IHES.
J. Moser,On commuting circle mappings and simultaneous Diophantine approximations, Math. Z.205 (1990), 105–121.
C. Roger,Méthodes homotopiques et cohomologiques en théorie de feuilletages, Thèse, Université de Paris XI, (1976).
I. Vaisman, “Cohomology and differential forms.” Dekker (1973).
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Arraut, J.L., dos Santos, N.M. Linear foliations ofT n . Bol. Soc. Bras. Mat 21, 189–204 (1991). https://doi.org/10.1007/BF01237364
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DOI: https://doi.org/10.1007/BF01237364