I'm sure people in the field know this, but I'm not in the field. Under what conditions (be they on the manifold or the map) is a diffeomorphism from a differentiable manifold $M$ to itself homotopic to the identity? Is this "usually" the case (by some definition of "usually")? And if it is homotopic to the identity, can we choose it to be a homotopy of diffeomorphisms?
Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
|
2
4
|
||||||||||||||
|
|
13
|
Your latter question the answer is generally no. A diffeomorphism $f : S^n \to S^n$ has homotopy class given by its degree $\pm 1$. But the homotopy-through-diffeomorphisms (usually called isotopy) classes are the group of exotic smooth structures on $S^{n+1}$ provided $n \geq 5$. There are large classes of manifolds for which the homotopy-classes of diffeomorphisms are reasonable. Hyperbolic $n$-manifolds for $n\geq 3$ have the property that homotopy-equivalences are homotopic to isometries. This is "Mostow rigidity". So homotopy-classes of diffeomorphisms are the same things as homotopy-classes of homotopy-equivalences in this case, which is $Out(\pi_1 M)$, since hyperbolic manifolds are $K(\pi,1)$-spaces. If you generate 3-manifolds via Heegaard splittings there is a sense in which most 3-manifolds are hyperbolic, so the above gives you an answer in one instance of your question. But in general there's not much known about the forgetful map $$\pi_0 Diff(M) \to \pi_0 HomEq(M)$$ Perhaps the largest obstruction to understanding this map is that we know so little about $\pi_0 Diff(M)$. In high dimensions surgery theory gives you some tools. |
|||
|
|
You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.
|
4
|
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. |
|||
|
