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.

[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.

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 X is a smooth or PL manifold, the mapping class group is interpreted to be the isotopy classes of diffeomorphisms or PL-automorphisms of the manifold.

If the mapping class group of X is finite then X is sometimes called rigid.

[edit] Examples

It is an easy exercise to prove:

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

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:

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

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. For more information on this topic see the Nielsen-Thurston classification theorem and the article on 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 Möbius strip, 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] 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 $H^1(\Sigma)\cong \mathbf{Z}^{2g}$ . Orientation-preserving maps are precisely those that act trivially on top cohomology $H^2(\Sigma) \cong \mathbf{Z}$ . $H 1 (Σ)$ 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:

$1 \to \mbox{Tor}(\Sigma) \to \mbox{MCG}(\Sigma) \to \mbox{Sp}(H^1(\Sigma)) \cong \mbox{Sp}_{2g}(\mathbf{Z}) \to 1$

One can extend this to

$1 \to \mbox{Tor}(\Sigma) \to \mbox{MCG}^*(\Sigma) \to \mbox{Sp}^{\pm}(H^1(\Sigma)) \cong \mbox{Sp}^{\pm}_{2g}(\mathbf{Z}) \to 1$

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

Mapping class group

From Wikipedia, the free encyclopedia

Contents

[edit] Definition

[edit] Examples

[edit] Torelli group

[edit] See also

[edit] References

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