Mapping Class Group - Definition

Definition

The term mapping class group has a flexible usage. Most often it is used in the context of a manifold M. The mapping class group of M is interpreted as the group of isotopy-classes of automorphisms of M. So if M is a topological manifold, the mapping class group is the group of isotopy-classes of homeomorphisms of M. If M is a smooth manifold, the mapping class group is the group of isotopy-classes of diffeomorphisms of M. Whenever the group of automorphisms of an object X has a natural topology, the mapping class group of X is defined as Aut(X)/Aut₀(X) where Aut₀(X) is the path-component of the identity in Aut(X). For topological spaces, this is usually the compact-open topology. In the low-dimensional topology literature, the mapping class group of X is usually denoted MCG(X), although it is also frequently denoted π₀(Aut(X)) where one substitutes for Aut the appropriate group for the category X is an object of. π₀ denotes the 0-th homotopy group of a space.

So in general, there is a short-exact sequence of groups:

Frequently this sequence is not split.

If working in the homotopy category, the mapping-class group of X is the group of homotopy-classes of homotopy-equivalences of X.

There are many subgroups of mapping class groups that are frequently studied. If M is an oriented manifold, Aut(M) would be the orientation-preserving automorphisms of M and so the mapping class group of M (as an oriented manifold) would be index two in the mapping class group of M (as an unoriented manifold) provided M admits an orientation-reversing automorphism. Similarly, the subgroup that acts trivially on the homology of M is called the Torelli group of M, one could think of this as the mapping class group of a homologically-marked surface.