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.

Motivation

Consider a topological space, that is, a space with some notion of closeness between points in the space. We can consider the set of homeomorphisms from the space into itself, that is, continuous functions with continuous inverses: functions which stretch and deform the space continuously without puncturing or breaking the space. This set of homeomorphisms can be thought of as a space itself. It forms a group under functional composition. We can also define a topology on this new space of homeomorphisms. The open sets of this new function space will be made up of sets of functions that map compact subsets K into open subsets U as K and U range throughout our original topological space, completed with their finite intersections (which must be open by definition of topology) and arbitrary unions (again which must be open). This gives a notion of continuity on the space of functions, so that we can consider continuous deformation of the homeomorphisms themselves: called homotopies. We define the mapping class group by taking homotopy classes of homeomorphisms, and inducing the group structure from the functional composition group structure already present on the space of homeomorphisms.

Definition

The term mapping class group has a flexible usage. Most often it is used in the context of a manifold M. The mapping class group of M is interpreted as the group of isotopy-classes of automorphisms of M. So if M is a topological manifold, the mapping class group is the group of isotopy-classes of homeomorphisms of M. If M is a smooth manifold, the mapping class group is the group of isotopy-classes of diffeomorphisms of M. Whenever the group of automorphisms of an object X has a natural topology, the mapping class group of X is defined as Aut(X)/Aut₀(X) where Aut₀(X) is the path-component of the identity in Aut(X)(Notice that in the compact-open topology, path components and isotopy classes coincide, i.e., two maps f,g are in the same path-component iff they are isotopic). For topological spaces, this is usually the compact-open topology. In the low-dimensional topology literature, the mapping class group of X is usually denoted MCG(X), although it is also frequently denoted π₀(Aut(X)) where one substitutes for Aut the appropriate group for the category X is an object of. π₀ denotes the 0-th homotopy group of a space.

So in general, there is a short-exact sequence of groups:

$1 \rightarrow {\rm Aut}_0(X) \rightarrow {\rm Aut}(X) \rightarrow {\rm MCG}(X) \rightarrow 1.$

Frequently this sequence is not split.

If working in the homotopy category, the mapping-class group of X is the group of homotopy-classes of homotopy-equivalences of X.

There are many subgroups of mapping class groups that are frequently studied. If M is an oriented manifold, Aut(M) would be the orientation-preserving automorphisms of M and so the mapping class group of M (as an oriented manifold) would be index two in the mapping class group of M (as an unoriented manifold) provided M admits an orientation-reversing automorphism. Similarly, the subgroup that acts trivially on the homology of M is called the Torelli group of M, one could think of this as the mapping class group of a homologically-marked surface.

Examples

Sphere

In any category (smooth, PL, topological, homotopy)

$\operatorname{MCG}(\mathbf{S}^2) \simeq {\mathbf Z}/2{\mathbf Z}$,

corresponding to maps of degree ±1.

Torus

In the homotopy category

${\rm MCG}(\mathbf{T}^n) \simeq {\rm SL}(n, {\mathbf Z}).$

This is because Tⁿ = (S¹)ⁿ is an Eilenberg-MacLane space.

For other categories if n ≥ 5, one has the following split-exact sequences:

In the category of topological spaces

$0\to \mathbf Z_2^\infty\to MCG(\mathbf{T}^n) \to GL(n,\mathbf Z)\to 0$

In the PL-category

$0\to \mathbf Z_2^\infty\oplus\binom n2\mathbf Z_2\to MCG (\mathbf{T}^n)\to GL(n,\mathbf Z)\to 0$

(⊕ representing direct sum). In the smooth category

$0\to \mathbf Z_2^\infty\oplus\binom n2\mathbf Z_2\oplus\sum_{i=0}^n\binom n i\Gamma_{i+1}\to MCG(\mathbf{T}^n)\to GL(n,\mathbf Z)\to 0$

where Γ_i are Kervaire-Milnor finite abelian groups of homotopy spheres and Z₂ is the group of order 2.

Surfaces

The mapping class groups of surfaces have been heavily studied, and are called Teichmüller modular groups. (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. For more information on this topic see the Nielsen–Thurston classification theorem and the article on Dehn twists. Every finite group is a subgroup of the mapping class group of a closed, orientable surface, moreover one can realize any finite group as the group of isometries of some compact Riemann surface.

Non-orientable surfaces

Some non-orientable surfaces have mapping class groups with simple presentations. For example, every homeomorphism of the real projective plane P²(R) is isotopic to the identity:

${\rm MCG}(\mathbf{P}^2(\mathbf{R})) = 1.$

The mapping class group of the Klein bottle K is:

${\rm MCG}(K)= \mathbf{Z}_2 \oplus \mathbf{Z}_2.$

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 N₃ has:

${\rm MCG(N_3)} = {\rm GL}(2, {\mathbf Z}).$

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.

3-Manifolds

Mapping class groups of 3-manifolds have received considerable study as well, and are closely related to mapping class groups of 2-manifolds. For example, any finite group can be realized as the mapping class group (and also the isometry group) of a compact hyperbolic 3-manifold.

Mapping-class groups of pairs

Given a pair of spaces (X,A) the mapping class group of the pair is the isotopy-classes of automorphisms of the pair, where an automorphism of (X,A) is defined as an automorphism of X that preserves A, i.e. f: X → X is invertible and f(A) = A.

Symmetry group of knot and links

If K ⊂ S³ is a knot or a link, the symmetry group of the knot (resp. link) is defined to be the mapping class group of the pair (S³, K). The symmetry group of a hyperbolic knot is known to be dihedral or cyclic, moreover every dihedral and cyclic group can be realized as symmetry groups of knots. The symmetry group of a torus knot is known to be of order two Z₂.

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, named after the Torelli theorem.

In the case of orientable surfaces, this is the action on first cohomology H¹(Σ) ≅ Z^2g. Orientation-preserving maps are precisely those that act trivially on top cohomology H²(Σ) ≅ Z. 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:

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

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.

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

Stable mapping class group

One can embed the surface $\Sigma_{g,1}$ of genus g and 1 boundary component into $\Sigma_{g+1,1}$ by attaching an additional hole on the end (i.e., gluing together $\Sigma_{g,1}$ and $\Sigma_{1,2}$), 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.

See also

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

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

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

External links

• Madsen-Weiss MCG Seminar; many references