nLab
mapping class group

Context

Manifolds and cobordisms

Group Theory

Contents

Idea

Given a (oriented) topological manifold XX, its mapping class group MCG(X)MCG(X) is the group of isotopy classes of (orientation preserving) homeomorphisms XXX\to X.

Often this is considered specifically for XX a Riemann surface with punctures in which case a central role is played by Dehn twists.

The mapping class group is of importance in many areas of geometry including study of Teichmüller spaces, of moduli spaces of surfaces, of automorphisms of free groups and in geometric and combinatorial group theory, hyperbolic geometry and so on. Some of the key contributors were Max Dehn, Jakob Nielsen, William Thurston, David Mumford. Recent proof of the related Mumford conjecture has been accomplished by Madsen and Weiss.

Definition

For Aut(X)\mathbf{Aut}(X) the automorphism group of the manifold formed in Euclidean topological geometry, hence equipped with its canonical structure of a topological group. Let furthermore Aut 0(X)Aut(X)\mathbf{Aut}_0(X) \hookrightarrow \mathbf{Aut}(X) be the inclusion of the connected component of the identity.

Then

MCG(X)Aut(X)/Aut 0(X) \mathbf{MCG}(X) \coloneqq \mathbf{Aut}(X)/\mathbf{Aut}_0(X)

is the corresponding coset space/quotient group. In other words, the mapping class group is the group of homeomorphism of XX onto itself, modulo isotopy.

This is a discrete group. Equivalently it is the group of connected components of Aut(X)\mathbf{Aut}(X). If XX is a smooth manifold, then the mapping class group is the group of connected components of the diffeomorphism group

MCG(X)=π 0(Diff(X)). MCG(X) = \pi_0(Diff(X)) \,.

If XX is a manifold with boundary X\partial X, then it is usual to consider automorphisms which restrict to the identity on the boundary.

Properties

For 2-dimensional surfaces

The mapping class group for 2-dimensional manifolds controls the moduli stack of complex curves.

The classifying spaces of mapping class groups for 2-dimensional manifolds may also be encoded combinatorially in the geometric realization of a category of ribbon graphs. See there for details.

One of the classical results is that the (oriented) mapping class group of the torus 2/ 2(S 1) 2\mathbb{R}^2/\mathbb{Z}^2 \cong (S^1)^2 is isomorphic to the special linear group SL 2()SL_2(\mathbb{Z}) (more generally, MCG( n/ n)SL n()MCG(\mathbb{R}^n/\mathbb{Z}^n) \cong SL_n(\mathbb{Z})). Certain generators called Dehn twists may be visualized as cutting a torus along a circle {a}×S 1\{a\} \times S^1 (or S 1×{b}S^1 \times \{b\}), thus producing a cylinder, then twisting one of the ends of the cylinder through 2π2\pi and reattaching the two ends.

Another example is a 2-disk with nn punctures. The group of diffeomorphisms (fixing the boundary pointwise) modulo isotopy is the braid group B nB_n.

The relation to the homotopy type of the diffeomorphism group is as follows:

Proposition

For Σ\Sigma a closed orientable surface, then the bare homotopy type of its diffeomorphism group is

  1. if Σ\Sigma is the sphere then

    Π(Diff(S 2)) Π(O(3)) MCG(S 2)×Π(SO(3)) 2×Π(SO(3)) \begin{aligned} \Pi(Diff(S^2)) & \simeq \Pi(O(3)) \\ & \simeq MCG(S^2)\times \Pi(SO(3)) \\ & \simeq \mathbb{Z}_2 \times \Pi(SO(3)) \end{aligned}
  2. if Σ\Sigma is the torus then

    Π(Diff(S 1×S 1)) MCG(S 1×S 1)×Π(S 1×S 1) GL 2()×B(×) \begin{aligned} \Pi(Diff(S^1 \times S^1)) & \simeq MCG(S^1 \times S^1)\times \Pi(S^1 \times S^1 ) \\ & \simeq GL_2(\mathbb{Z}) \times B(\mathbb{Z} \times\mathbb{Z}) \end{aligned}
  3. in all other cases all higher homotopy groups vanish:

    Π(Diff(Σ))MCG(Σ) \Pi(Diff(\Sigma)) \simeq MCG(\Sigma)

The first statement is due to (Smale 58), see also at sphere eversion. The second and third are due to (Earle-Eells 67, Gramain 73).

See (Hatcher 12) for review.

Rational cohomology

The ordinary cohomology with rational coefficients of the delooping of the stable mapping class group of 2-dimensional manifolds (hence essentially the orbifold cohomology of the moduli stack of complex curves) is the content of Mumford's conjecture, proven in (Madsen-Weiss 02).

References

Surveys include

See also

C.J. Earle, J. Eells, The diffeomorphism group of a compact Riemann surface, Bulletin of the American Mathematical Society 73(4) 557–559, 1967