For a manifold $M$, define the mapping class group $Mod(M)$ to be the set of self-diffeomorphisms of $M$, modulo isotopy. In symbols, $Mod(M) = \pi\_0 Diff(M)$. Of course, every self-diffeomorphism gives an automorphism of $\pi\_1(M)$, well-defined up to conjugacy because of basepoint issues. Thus we have a homomorphism $\sigma: Mod(M) \to Out ( \pi\_1 M)$.


When $M$ is a surface, the Dehn-Nielsen-Baer theorem says $\sigma: Mod(M) \to Out ( \pi\_1 M)$ is an isomorphism. My question is: what can be said in higher dimensions?


Assuming $M$ is a $K(\pi, 1)$, one can identify $Out ( \pi\_1 M)$ with the set of homotopy classes of homotopy equivalences of $M$. Through this lens, injectivity of $\sigma$ is the question of whether two homotopic diffeomorphisms need to be isotopic. Surjectivity of $\sigma$ is the question of whether every homotopy equivalence is homotopic to a self-diffeomorphism.


From a naive point of view, both injectivity and surjectivity seem hard. What is known about them?



Generalizations of Dehn-Nielsen-Baer
In general it's not injective, nor surjective. It's just a map. 


For example, if $M = S^n$, the sphere. $\pi\_0 Diff(M)$ is the group of homotopy $(n+1)$-spheres, but $Out(\pi\_1 M)$ is trivial. So you have surjectivity but not injectivity provided exotic spheres exist in that dimension. There's lots of much more complicated examples of this type, for example $M = S^1 \times D^n$, this goes back to work of Farrell, Hatcher, Quillen, Igusa.


I think you mean to formulate the question a little differently since manifolds can have orientiation-reversing diffeomorphisms and you rarely "capture" that by $Out(\pi\_1)$. For example, $\pi\_0 Diff(S^2) \simeq \mathbb Z\_2$ yet $Out(\pi\_1 S^2)$ is trivial. 


But similarly, $\pi\_0 Diff(S^2 \times S^1) \simeq \mathbb Z\_2^2$ and $Out(\pi\_1 S^2 \times S^1)$ is $\mathbb Z\_2$. But even in the orientation-preserving case, there's no isomorphism. 


edit: okay if you're specifically interested in $K(\pi,1)$ spaces, there's the computation of Hatcher -- $\pi\_0 Diff( (S^1)^n)$ is an extension of $GL\_n \mathbb Z$ by:


$$ \mathbb Z\_2^\infty\oplus\binom n2\mathbb Z\_2\oplus\sum\_{i=0}^n\binom n i\Gamma\_{i+1} $$


That's an infinite direct-sum of $\mathbb Z\_2$'s together with a direct sum of many groups of exotic spheres. See the [Wikipedia page](http://en.wikipedia.org/wiki/Mapping_class_group). 


