# Mapping class group

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

[itex]{\rm Homeo}(X)[itex]

be the homeomorphism group of X. Let

[itex]{\rm Homeo}_0(X)[itex]

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

[itex]{\rm MCG}(X) = {\rm Homeo}(X) / {\rm Homeo}_0(X)[itex]

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

[itex]1 \rightarrow {\rm Homeo}_0(X) \rightarrow {\rm Homeo}(X) \rightarrow {\rm MCG}(X) \rightarrow 1[itex]

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 [itex]{\rm Homeo}^+(X)[itex]. Here convention dictates that the group defined in the second paragraph be called the extended mapping class group.

## Examples

An easy exercise is to show that:

[itex] {\rm MCG}(S^1) = {\mathbb Z}/2{\mathbb Z}. [itex]

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

[itex] {\rm MCG}(T^n) = {\rm GL}(n, {\mathbb Z}). [itex]

Finally, the mapping class groups of surfaces have been heavily studied. (Note the special case of [itex] {\rm MCG}(T^2)[itex] 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. The mapping class group of any closed, orientable surface can be generated by Dehn twists.

Another fine example is the mapping class group of the Klein bottle [itex] K [itex]. This is

[itex] {\rm MCG}(K)={\mathbb Z}/2{\mathbb Z}+{\mathbb Z}/2{\mathbb Z}. [itex]

Determined topologically by W.B.R. Lickorish in 1963 and algebraically by H. Torriani in 1987.

## References

For more information on this last topic 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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