Connexité des actions lisses de Z^2 sur l'intervalle
Résumé: Smooth actions of Z^2 on the segment [0,1], seen as holonomy
representations, describe codimension one foliations on the thick torus
T^2 x [0,1] transverse to the second factor and having the boundary tori
as leaves. As it turns out, studying the space of such actions is
important to understand the space of codimension 1 foliations on ANY
3-manifold.
This motivates the following result, obtained in a joint work with C.
Bonatti: the space of smooth orientation preserving actions of Z^2 on
[0,1], naturally identified to the space of pairs of commuting
increasing diffeomorphisms of [0,1], is connected.
We don't know, however, if it is path-connected. Of course, deforming a
given pair into any other one is not difficult, the space of smooth
increasing diffeomorphisms of the segment being contractible. But we
will see, appealing to some classical results of Szekeres and Kopell,
that the commutativity condition rigidifies the problem immensely. We
will then explain how to take advantage of this rigidity to prove our
result, and why our method doesn't yield path-connectedness.
