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In mathematics, in the sub-field of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.

Contents 1 Definition 2 Examples 3 See also 4 References on mapping class groups of surfaces

Suppose that X is a topological space. Let

Homeo(X)

be the group of self-homeomorphisms of X. Let

Homeo₀(X)

be the subgroup of Homeo(X) consisting of all homeomorphisms isotopic to the identity map on X. It is easy to verify that Homeo₀(X) is in fact a subgroup and is normal. The factor group

MCG(X) = Homeo(X) / Homeo₀(X)

is the mapping class group of X. Thus there is a natural short exact sequence:

As usual, there is interest in the spaces where this sequence splits.

Some mathematicians, when X is an orientable manifold, restrict attention to orientation-preserving homeomorphisms Homeo ⁺ (X). Here convention dictates that the group defined in the second paragraph be called the extended mapping class group, MCG*(X).

If the mapping class group of X is finite then X is sometimes called rigid.

It is an easy exercise to prove:

The mapping class group may also be infinite. Taking Tⁿ to be the n-dimensional torus we find that the extended mapping class group is isomorphic to the general linear group over the integers:

The mapping class groups of surfaces have been heavily studied. (Note the special case of MCG ^* (T²) above.) This is perhaps due to their strange similarity to higher rank linear groups as well as many applications, via surface bundles, in Thurston's theory of geometric three-manifolds. We note that the non-extended mapping class group of any closed, orientable surface can be generated by Dehn twists.

Some non-orientable surfaces have mapping class groups with simple presentations. For example, every homeomorphism of the real projective plane is isotopic to the identity:

The mapping class group of the Klein bottle K is:

The four elements are the identity, a Dehn twist on the two-sided curve which does not bound a Mobius band, the y-homeomorphism of Lickorish, and the product of the twist and the y-homeomorphism. It is a nice exercise to show that the square of the Dehn twist is isotopic to the identity.

We also remark that the closed genus three non-orientable surface N₃ has:

This is because the surface has a unique one-sided curve that, when cut open, yields a once-holed torus. This is discussed in a paper of Martin Scharlemann.

• Braid groups, the mapping class groups of punctured discs.
• Homotopy groups.
• Homeotopy groups.

• Braids, Links, and Mapping Class Groups by Joan Birman.
• Automorphism of surfaces after Nielsen and Thurston by Andrew Casson and Steve Bleiler.
• "Mapping Class Groups" by Nikolai V. Ivanov in the Handbook of Geometric Topology.