Manifolds and cobordisms
Cohomology and Extensions
Given a (oriented) topological manifold , its mapping class group is the group of isotopy classes of (orientation preserving) homeomorphisms .
Often this is considered specifically for a Riemann surface with punctures in which case a central role is played by Dehn twists.
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.
is the corresponding coset space/quotient group. In other words, the mapping class group is the group of homeomorphism of onto itself, modulo isotopy.
This is a discrete group. Equivalently it is the group of connected components of . If is a smooth manifold, then the mapping class group is the group of connected components of the diffeomorphism group
If is a manifold with boundary , then it is usual to consider automorphisms which restrict to the identity on the boundary.
For 2-dimensional surfaces
The mapping class group for 2-dimensional manifolds controls the moduli stack of complex curves.
The classifying spaces of mapping class groups for 2-dimensional manifolds may also be encoded combinatorially in the geometric realization of a category of ribbon graphs. See there for details.
One of the classical results is that the (oriented) mapping class group of the torus is isomorphic to the special linear group (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 relation to the homotopy type of the diffeomorphism group is as follows:
For a closed orientable surface, then the bare homotopy type of its diffeomorphism group is
if is the sphere then
if is the torus then
in all other cases all higher homotopy groups vanish:
The first statement is due to (Smale 58), see also at sphere eversion. The second and third are due to (Earle-Eells 67, Gramain 73).
See (Hatcher 12) for review.
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).
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wikipedia: mapping class group
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