The mapping class group is of importance in many areas of geometry including study of Teichmüller spaces, of moduli spaces of surfaces, of automorphisms of free groups and in geometric and combinatorial group theory, hyperbolic geometry and so on. Some of the key contributors were Max Dehn, Jakob Nielsen, William Thurston, David Mumford. Recent proof of the related Mumford conjecture has been accomplished by Madsen and Weiss.
For the automorphism group of the manifold formed in Euclidean topological geometry, hence equipped with its canonical structure of a topological group. Let furthermore be the inclusion of the connected component of the identity.
If is a manifold with boundary , then it is usual to consider automorphisms which restrict to the identity on the boundary.
The mapping class group for 2-dimensional manifolds controls the moduli stack of complex curves.
One of the classical results is that the mapping class group of the torus is isomorphic to (more generally, ). Certain generators called Dehn twists may be visualized as cutting a torus along a circle (or ), thus producing a cylinder, then twisting one of the ends of the cylinder through and reattaching the two ends.
Another example is a 2-disk with punctures. The group of diffeomorphisms (fixing the boundary pointwise) modulo isotopy is the braid group .
The ordinary cohomology with rational coefficients of the delooping of the stable mapping class group of 2-dimensional manifolds (hence essentially the orbifold cohomology of the moduli stack of complex curves) is the content of Mumford's conjecture, proven in (Madsen-Weiss 02).
Massyuyeau, A short introduction to mapping class groups (pdf)
Benson Farb, Dan Margalit, A primer on mapping class groups, draft, web
Nikolai V. Ivanov. Mapping class groups. In Handbook of geometric topology, pages 523–633. North-Holland, Amsterdam, 2002; 1998 draft: ps
wikipedia: mapping class group
John Harer, The second homology group of the mapping class group of an orientable surface, Invent. Math., 72(2):221–239, 1983; The cohomology of the moduli space of curves in: Theory of moduli (Montecatini Terme, 1985), Lecture Notes in Math. 1337, p. 138–221. Springer, Berlin, 1988.
Robert C. Penner, The decorated Teichmüller space of punctured surfaces, Commun. Math. Phys. 113 (2) (1987) 299–339. MR89h:32044; A construction of pseudo-Anosov homeomorphisms, Trans. Amer. Math. Soc. 310 (1):179–197, 1988.
Jakob Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. I-III, Acta Math. 50 (1927), no. 1, 189–358, MR1555256, doi; Acta Math. 53 (1929), no. 1, 1–76, MR1555290, doi; Acta Math. 58 (1932), no. 1, 87–167, MR1555345, doi
David Mumford, Abelian quotients of the Teichmüller modular group, J. Analyse Math., 18:227–244, 1967.
S. Morita, Introduction to mapping class groups of surfaces and related groups, in: Handbook of Teichmüller theory (A. Papadopoulos, editor), vol. I, EMS Publishing House, Zürich, 2007, 353–386.