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 and let

Homeo 0 (X)

be the subset 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 in Homeo(X). The factor group

MCG(X) = Homeo(X) / Homeo 0 (X)

is the mapping class group of X; its elements are the isotopy classes of self-homeomorphisms 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.

If we introduce a suitable topology on Homeo(X) (so that isotopies become continuous paths and Homeo(X) becomes a topological group), then Homeo₀(X) is the set of elements of Homeo(X) path-connected to the identity. The mapping class group is then the 0th homotopy group, $MCG(X) = π 0 (Homeo(X))$ .

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). (Compare with notation S*L for the extended special linear group of matrices with determinant ±1.) A technical note is that X need only be orientable (able to be oriented), not oriented to define the orientation-preserving maps.

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

[edit] Sphere

It is an easy exercise to prove that:

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

corresponding to degree +1 and $- 1$ . These are the only degrees that are invertible on the fundamental class (i.e., for algebraic reasons), and can clearly be represented by homeomorphisms (the identity and a reflection); the exercise is that two homeomorphisms of the same degree are not just homotopic, but isotopic. This is a "homeomorphism reduces to homotopy reduces to algebra" result.

[edit] Torus

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:

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

Further, the oriented mapping class group is ${\rm SL}(n, {\mathbb Z}).$ These can be realized as linear maps on $\mathbb{R}^n$ , the universal cover of the torus – the integer coefficients correspond to preserving the standard lattice $\mathbb{Z}^n$ , and thus descending to the quotient – and thus the group of homeomorphisms splits as the semidirect product $\mathrm{Homeo}(T^n) \cong \mathrm{Homeo}_0(T^n) \rtimes \mathrm{GL}(n,\mathbb{Z}).$

More canonically, given a maximal lattice in a vector space $Λ < V,$ the extended mapping class group of the quotient torus $V / Λ$ is naturally the group of linear maps of V preserving this lattice, $GL(Λ, V).$

At the level of homotopy and homology, the mapping class group of the torus can be identified as the action on the first homology (or equivalently, first cohomology, or on the fundamental group, as these are all naturally isomorphic; note also that the first cohomology group generates the cohomology algebra):

$\mathrm{MCG}^*(T^n) = \mathrm{Aut}(\pi_1(X)) = \mathrm{Aut}(\mathbb{Z}^n) = \mathrm{GL}(n,\mathbb{Z}).$

Since the torus is an Eilenberg-MacLane space K(G, 1), its homotopy equivalences, up to homotopy, can be identified with automorphisms of the fundamental group; that this agrees with the mapping class group reflects that all homotopy equivalences can be realized by homeomorphisms, and that homotopic homeomorphisms are in fact isotopic (connected through homeomorphisms, not just through homotopy equivalences). More tersely, the map $\mathrm{Homeo}(T^n) \to \mathrm{SHE}(T^n)$ is 1-connected (isomorphic on path-components, onto fundamental group).

This is another "homeomorphism reduces to homotopy reduces to algebra" result.

[edit] Surfaces

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.

[edit] Non-orientable surfaces

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 (because all elements are isotopic, hence homotopic to the identity, which acts trivially, and action on (co)homology is invariant under homotopy). The kernel of this action is the Torelli group.

In the case of orientable surfaces, this is the action on first cohomology $\scriptstyle H^1(\Sigma)\cong \mathbb{Z}^{2g}$ . Orientation-preserving maps are precisely those that act trivially on top cohomology $\scriptstyle H^2(\Sigma) \cong \mathbb{Z}$ . $\scriptstyle H^1(\Sigma)$ 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:

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

One can extend this to

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

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

Note that for the torus (genus 1) the map to the symplectic group is an isomorphism, and the Torelli group vanishes.

[edit] Stable mapping class group

One can embed the surface $Σ g,1$ of genus g and 1 boundary component into $Σ g + 1,1$ by attaching an additional hole on the end (connected sum with $Σ g,2$ ), and thus automorphisms of the small surface fixing the boundary extend to the larger surface. Taking the direct limit of these groups and inclusions yields the stable mapping class group, whose rational cohomology ring was conjectured by David Mumford (one of conjectures called the Mumford conjectures). The integral (not just rational) cohomology ring was computed in 2002 by Madsen and Weiss, proving Mumford's conjecture.

[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

[edit] Stable mapping class group

The stable moduli space of Riemann surfaces: Mumford's conjecture, by Ib Madsen and Michael S. Weiss, 2002

Published as: The stable moduli space of Riemann surfaces: Mumford's conjecture, by Ib Madsen and Michael S. Weiss, 2007, Annals of Mathematics

[edit] External links

Madsen-Weiss MCG Seminar; many references

Mapping class group

From Wikipedia, the free encyclopedia

Contents

[edit] Definition

[edit] Examples

[edit] Sphere

[edit] Torus

[edit] Surfaces

[edit] Non-orientable surfaces

[edit] Torelli group

[edit] Stable mapping class group

[edit] See also

[edit] References

[edit] Stable mapping class group

[edit] External links

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