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.

To be more precise, suppose that X is a topological space. Let

<math>{\rm Homeo}(X)</math>

be the group of self homeomorphisms of X. Let

<math>{\rm Homeo}_0(X)</math>

be the subgroup of <math>{\rm Homeo}(X)</math> consisting of all homeomorphisms isotopic to the identity map on X. It is easy to verify that <math>{\rm Homeo}_0(X)</math> is in fact a subgroup and is normal. The factor group

<math>{\rm MCG}(X) = {\rm Homeo}(X) / {\rm Homeo}_0(X)</math>

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

<math>1 \rightarrow {\rm Homeo}_0(X) \rightarrow {\rm Homeo}(X) \rightarrow {\rm MCG}(X) \rightarrow 1</math>

As usual, there is interest in the spaces where this sequence splits. If the mapping class group of X is finite then X is sometimes called rigid.

Some mathematicians, when X is an orientable manifold, restrict attention to orientation-preserving homeomorphisms <math>{\rm Homeo}^+(X)</math>. Here convention dictates that the group defined in the second paragraph be called the extended mapping class group.

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Examples

An easy exercise is to show that:

<math> {\rm MCG}(S^1) = {\mathbb Z}/2{\mathbb Z}. </math>

For manifolds of dimension two or higher the mapping class group is often infinite. Generalizing the above, for the n-torus we have:

<math> {\rm MCG}(T^n) = {\rm GL}(n, {\mathbb Z}). </math>

The mapping class groups of surfaces have been heavily studied. (Note the special case of <math> {\rm MCG}(T^2)</math> 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 (orientation preserving) mapping class group of any closed, orientable surface can be generated by Dehn twists.

A few mapping class groups of non-orientable surface have simple presentations:

In the projective plane <math> {\mathbb RP}^2</math>, every self-homeomorphism is isotopic to the identity so

<math> {\rm MCG}({\mathbb RP}^2) = 1. </math>

The mapping class group of the Klein bottle <math> K </math> is:

<math> {\rm MCG}(K)={\mathbb Z}/2{\mathbb Z}\oplus{\mathbb Z}/2{\mathbb Z}. </math>

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 <math>N_3</math> also has <math>{\rm MCG(N_3)}</math> equal to

<math> {\rm GL}(2, {\mathbb Z}). </math>

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.

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References

For more information relating to the mapping class groups of surface one should consult the book by Andrew Casson and Steve Bleiler entitled Automorphisms of surfaces after Nielsen and Thurston. The special case where X is a punctured disk is discussed by Joan Birman in Braids, Links, and Mapping Class Groups.

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