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.

Contents

Definition

Examples

See also

References on mapping class groups of surfaces

Definition

Suppose that ''X'' is a topological space. Let
:$\{$
m Homeo}(X)
be the group of self-homeomorphisms of ''X''. Let
:$\{$
m Homeo}_0(X)
be the subgroup of $\{$
m Homeo}(X) consisting of all homeomorphisms isotopic to the identity map on ''X''. It is easy to verify that $\{$
m Homeo}_0(X) is in fact a subgroup and is normal. The factor group
:$\{$
m MCG}(X) = {
m Homeo}(X) / {
m Homeo}_0(X)
is the ''mapping class group'' of ''X''. Thus there is a natural short exact sequence:
:$1$
ightarrow {
m Homeo}_0(X)
ightarrow {
m Homeo}(X)
ightarrow {
m MCG}(X)
ightarrow 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 $\{$
m 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.

Examples

It is an easy exercise to prove:
:$\{$
m MCG^
★ }(S^2) = {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:
:$\{$
m MCG^
★ }(T^n) = {
m GL}(n, {mathbb Z}).
The mapping class groups of surfaces have been heavily studied. (Note the special case of $\{$
m 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:
:$\{$
m MCG}({mathbb RP}^2) = 1.
The mapping class group of the Klein bottle $K$ is:
:$\{$
m 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:
:$\{$
m MCG(N_3)} = {
m 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.

See also

★ Braid groups, the mapping class groups of punctured discs.

★ Homotopy groups.

★ Homeotopy groups.

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''.

★ ''A Primer on Mapping Class Groups'' by Benson Farb and Dan Margalit