TITLE: Compact manifolds with big mapping class group
QUESTION [14 upvotes]: I was wondering if compact surfaces were the only compact manifolds with a "big" or "complicated" mapping class group. 
Are there higher dimensional manifolds (which are not in some way reducible to a surface) that have an infinite mapping class group? What kind of group can arise as the mapping class group of a compact manifold?
Thanks for your attention :)

REPLY [12 votes]: In 
 Infinitesimal computations in topology  Sullivan shows in Theorem 13.3 that if $M$ is a simply-connected manifold of dimension $>5$, then $\pi_0(\mathrm{Diff\,} M)$ is commensurable to an arithmetic group (see p.295 for definitions of an arithmetic group). 
The group of homotopy classes of homotopy self-equivalences is also commensurable to an arithmetic one (Theorem 10.3)
In Corollary 13.3  Sullivan explains how the forgetful map between the two is largely controlled by the rational Pontryagin class of the manifold (which is preserved by diffeomorphisms by not by homotopy equivalences).