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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?

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Any hint on which manifold you are interested in? Otherwise, it will be very hard to give a specific answer. Most manifolds admit many self-diffeomorphisms that are not homotopic to the identity. And if a diffeomorphism is homotopic to the identity, it does not mean that it is isotopic ("homotopy of diffeomorphisms") to the identity. –  Johannes Ebert Nov 7 '10 at 20:15
    
I'm talking about general differentiable manifolds. The only obstruction I could find thinking about examples is that for orientable differential manifolds the diffeomorphism has to be orientation preserving. If you can claim successfully that "usually" it's not the case that a diffeomorphism is homotopic to the identity, that would be an answer too. –  James D. Taylor Nov 7 '10 at 20:22
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Usually a diffeomorphism is not isotopic to the identity. Some buzzwords here are "mapping class group". Also look up "Dehn twist". For a simple example of Dehn twist, take a torus (genus 1 surface), then imagine cutting along a meridian (so you get a tube), and then twist one end through 360 degrees and glue the ends back together. This gives a nontrivial element in the mapping class group (the twist is not isotopic to the identity), and I believe this group is identified with SL_2(Z). However, I'm not an expert on this stuff. –  Todd Trimble Nov 8 '10 at 1:01
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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.

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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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could you cite here clearly the reference for the fact ''For surfaces it is true that a diffeomorphism which is homotopic to id is an isotopy (see Gabai and many others);'. thank. –  vu viet Feb 19 '13 at 16:23
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