Espaces différentiables singuliers et corps de nombres algébriques

Espaces différentiables singuliers et corps de nombres algébriques

July 15, 1990

In this paper we study the diffeology of the space of leaves of codimension 1 linear foliations of tori Tn and their classification. It is a good example of how diffeology deals with objects which are not manifolds and which are inaccessible to topology.

We consider the quotient of a torus Tn = Rn/Zn, by an hyperplane H totally irrational, i.e.
H ∩Z = {0}.
Let w = (w1...wn) be the director 1-form of H, that is H = ker(w). The quotient TH = Tn/H is equipped with the quotient diffeology. First of all, we show that the group of the connected components of the group of diffeomorphisms Diff(TH) is isomorphic to the stabilizer of H, for the action of GL(n,Z)
π0(Diff(TH)) ~ StabGL(n,Z)(H).
Then, we show that the group StabGL(n,Z)(H) is isomorphic to the group of units of some order of a characteristic field KH of algebraic number, defined as follows: compute this group. We introduce the set of real numbers KH defined by :
KH = { k ∈ R | k.EH ⊂ EH } where EH = w(Qn) ⊂ R
EH is a Q-vector space generated by the director coefficients wi (i=1...n) of H. KH is the set of numbers which stabilize EH by multiplication. Then, thanks to the Dirichlet theorem we conclude that:
π0(Diff(TH)) ~ Zr+s-1
where r is the number of real places of KH and s the number of complex places. Finally, we show that the group of unitits of any order of any finite extension of Q can be realized as the π0(Diff(TH)) for some totally irrational linear codimension 1 foliation of some torus Tn.

Published in Annales de l’Institut Fourier, tome 40, N° 3, pages 723-737, 1990.