# When are diffeomorphisms from a manifold to itself homotopic to the identity?

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
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

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