Mapping class group

In any category (smooth, PL, topological, homotopy)[2]

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

corresponding to maps of degree ±1.

In the homotopy category

${\displaystyle \operatorname {MCG} (\mathbf {T} ^{n})\simeq \operatorname {GL} (n,\mathbb {Z} ).}$

This is because the n-dimensional torus ${\displaystyle \mathbf {T} ^{n}=(S^{1})^{n}}$ is an Eilenberg–MacLane space.

For other categories if ${\displaystyle n\geq 5}$,[3] one has the following split-exact sequences:

In the category of topological spaces

${\displaystyle 0\to \mathbb {Z} _{2}^{\infty }\to \operatorname {MCG} (\mathbf {T} ^{n})\to \operatorname {GL} (n,\mathbb {Z} )\to 0}$

In the PL-category

${\displaystyle 0\to \mathbb {Z} _{2}^{\infty }\oplus {\binom {n}{2}}\mathbb {Z} _{2}\to \operatorname {MCG} (\mathbf {T} ^{n})\to \operatorname {GL} (n,\mathbb {Z} )\to 0}$

(⊕ representing direct sum). In the smooth category

${\displaystyle 0\to \mathbb {Z} _{2}^{\infty }\oplus {\binom {n}{2}}\mathbb {Z} _{2}\oplus \sum _{i=0}^{n}{\binom {n}{i}}\Gamma _{i+1}\to \operatorname {MCG} (\mathbf {T} ^{n})\to \operatorname {GL} (n,\mathbb {Z} )\to 0}$

where ${\displaystyle \Gamma _{i}}$ are the Kervaire–Milnor finite abelian groups of homotopy spheres and ${\displaystyle \mathbb {Z} _{2}}$ is the group of order 2.

The mapping class groups of surfaces have been heavily studied, and are sometimes called Teichmüller modular groups (note the special case of ${\displaystyle \operatorname {MCG} (\mathbf {T} ^{2})}$ above), since they act on Teichmüller space and the quotient is the moduli space of Riemann surfaces homeomorphic to the surface. These groups exhibit features similar both to hyperbolic groups and to higher rank linear groups. They have many applications in Thurston's theory of geometric three-manifolds (for example, to surface bundles). The elements of this group have also been studied by themselves: an important result is the Nielsen–Thurston classification theorem, and a generating family for the group is given by Dehn twists which are in a sense the "simplest" mapping classes. Every finite group is a subgroup of the mapping class group of a closed, orientable surface,;[4] in fact one can realize any finite group as the group of isometries of some compact Riemann surface (which immediately implies that it injects in the mapping class group of the underlying topological surface).

Some non-orientable surfaces have mapping class groups with simple presentations. For example, every homeomorphism of the real projective plane ${\displaystyle \mathbf {P} ^{2}(\mathbb {R} )}$ is isotopic to the identity:

${\displaystyle \operatorname {MCG} (\mathbf {P} ^{2}(\mathbb {R} ))=1.}$

The mapping class group of the Klein bottle K is:

${\displaystyle \operatorname {MCG} (K)=\mathbb {Z} _{2}\oplus \mathbb {Z} _{2}.}$

The four elements are the identity, a Dehn twist on a 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 (the connected sum of three projective planes) has:

${\displaystyle \operatorname {MCG} (N_{3})=\operatorname {GL} (2,\mathbb {Z} ).}$

This is because the surface N has a unique class of one-sided curves such that, when N is cut open along such a curve C, the resulting surface ${\displaystyle N\setminus C}$ is a torus with a disk removed. As an unoriented surface, its mapping class group is ${\displaystyle \operatorname {GL} (2,\mathbb {Z} )}$. (Lemma 2.1[5]).

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.[6]

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.

• Madsen, Ib; Weiss, Michael (2007). "The stable moduli space of Riemann surfaces: Mumford's conjecture". Annals of Mathematics. 165 (3): 843–941. arXiv:math/0212321. CiteSeerX 10.1.1.236.2025. doi:10.4007/annals.2007.165.843. JSTOR 20160047. S2CID 119721243.