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?