Mapping class group

From Wikipedia, the free encyclopedia

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 Torelli group 4 See also 5 References

[edit] Definition

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.

When X is an orientable manifold, it is often useful to restrict one's attention to orientation-preserving homeomorphisms Homeo ⁺ (X). Here, convention dictates that MCG(X) denotes the orientation-preserving version, and the group defined above is called the extended mapping class group and denoted 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:

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.

[edit] Torelli group

Notice that there is an induced action of the mapping class group on the homology (and cohomology) of the space X. This is because (co)homology is functorial and Homeo₀ acts trivially. The kernel of this action is the Torelli group.

In the case of orientable surfaces, this is the action on first cohomology . Orientation-preserving maps are precisely those that act trivially on top cohomology . H¹(Σ) has a symplectic structure, coming from the cup product; since these maps are automorphisms, and maps preserve the cup product, the mapping class group acts as symplectic automorphisms, and indeed all symplectic automorphisms are realized, yielding the short exact sequence:

One can extend this to

The symplectic group is well-understood. Hence understanding the algebraic structure of the mapping class group often reduces to questions about the Torelli group.

[edit] See also

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

[edit] References

• Braids, Links, and Mapping Class Groups by Joan Birman.
• Automorphisms 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