TITLE: Mapping class groups in higher dimensions
QUESTION [7 upvotes]: I am in the process of learning about Mapping class groups. At this point it seems like most of what I've read involves very low dimensional (surfaces and 3-manifolds) applications.
I was wondering if they were studied (or arise naturally) in higher dimensional settings?
In particular, any references to their uses in homotopy theory would be appreciated.

REPLY [6 votes]: In high dimensions there are several variants that are all distinct (which for surfaces they all agree).  There's mapping class groups in the "homotopy category" meaning the homotopy-classes of homotopy equivalences of a topological space, with composition giving the group structure.  This is a "core" object of study of classical algebraic topology.  In the topological/pl/smooth categories there are isotopy classes of homeomorphisms/pl automorphisms/diffeomorphisms of a manifold.  The smooth category gets quite a bit of attention -- for example the smooth category mapping class group of $S^n$ (if you restrict to orientation-preserving diffeomorphisms) is the group of homotopy $(n+1)$-spheres, provided $n \geq 5$.   There has been some work on stable high-dimensional mapping class groups by people like Giansiracusa (Swansea).
I have to head out but I can add more later.
The Giansiracusa reference is this: http://www.arxiv.org/abs/math.gt/0510599
Modulo some qualifiers the statement is that the stable mapping class group of a 4-manifold is the automorphisms of homology that preserve the intersection form.
Mapping class groups of a products of circles $(S^1)^n$ in the topological, PL and smooth categories were computed by Hatcher in his "Higher simple homotopy theory" paper.
Is there anything in particular you're interested in?
edit: In a little self-plug, David Gabai and I recently proved a result that contrasts heavily with the stability results referenced in the Giansiracusa paper.  Specifically, we show that the mapping class group (smooth diffeomorphisms) of $S^1 \times D^3$ and $S^1 \times S^3$ are not finitely generated. Tadayuki Watanabe has also been able to prove this result for $S^1 \times D^3$ using fairly different techniques.