Answer by Gael Meigniez for A local transitivity property of the automorphism group of a foliated manifold

2010-11-08T11:05:28Z

this will very rarely be true! remember that the leaves without holonomy are always generic. On the other hand the union of the leaves with holonomy is often dense. In that case, since any automorphism of the foliation will preserve holonomy, the will be no automorphism of your kind.

Answer by Gael Meigniez for When are diffeomorphisms from a manifold to itself homotopic to the identity?

2010-11-08T10:52:32Z

In differential topology, there is the notion of "isotopy". A isotopy of M is a self-diffeomorphism which is linked to the identity by a 1-parameter family of self-diffeomorphisms. The set of isotopies is denoted Diff0(M), since it is the neutral connected component in the topological group Diff(M). The quotient group Diff(M)/Diff0(M) is called the Mapping class group (MPG) and has been much studied for surfaces.

MPG(any surface) is generated by the Dehn twists.

For the 2-torus, MPG(T2)=SL2(Z). Indeed any matrix A in SL(2,Z) is a diffeomorphism of R^2 preserving Z^2, and thus a diffeomorphism of T^2=R^2/Z^2. The reason why it is not isotopic to the identity for A \neq id is that it is not even homotopic to the identity, since its action on \pi_1(T^2)=Z^2 is also A itself!

MPG(the disk minus k points) is the braid group on k braids.

For surfaces it is true that a diffeomorphism which is homotopic to id is an isotopy (see Gabai and many others); but in the genaral case it needs not be.

In general, a self-diffeomorphism f of M has to act by the identity on the homotopy groups and homology groups of M to be homotopic to the identity, and you will easily make many counterexamples.

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Answer by Gael Meigniez for A local transitivity property of the automorphism group of a foliated manifold

Answer by Gael Meigniez for When are diffeomorphisms from a manifold to itself homotopic to the identity?