Mapping class group

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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.

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

[edit] Definition

Suppose that X is a topological space. Let

Homeo(X)

be the group of self-homeomorphisms of X. Let

Homeo 0 (X)

be the subgroup of $Homeo(X)$ consisting of all homeomorphisms isotopic to the identity map on X. It is easy to verify that $Homeo 0 (X)$ is in fact a subgroup and is normal. The factor group

MCG(X) = Homeo(X) / Homeo 0 (X)

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

$1 \rightarrow {\rm Homeo}_0(X) \rightarrow {\rm Homeo}(X) \rightarrow {\rm MCG}(X) \rightarrow 1$

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.

[edit] Examples

It is an easy exercise to prove:

${\rm MCG^*}(S^1) = {\mathbb Z}/2{\mathbb Z}.$

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

${\rm MCG^*}(T^n) = {\rm GL}(n, {\mathbb Z}).$

The mapping class groups of surfaces have been heavily studied. (Note the special case of $MCG * (T 2)$ 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 ${\mathbb RP}^2$ is isotopic to the identity:

${\rm MCG}({\mathbb RP}^2) = 1.$

The mapping class group of the Klein bottle $K$ is:

${\rm MCG}(K)={\mathbb Z}/2{\mathbb Z}\oplus{\mathbb Z}/2{\mathbb Z}.$

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 $N 3$ has:

${\rm MCG(N_3)} = {\rm GL}(2, {\mathbb Z}).$

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.

[edit] See also

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

[edit] References on mapping class groups of surfaces

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.

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Categories: Geometric topology | Homeomorphisms