Mapping Class Groups

Introduction

The mapping class group of a surface is the group of the isotopy classes of its orientation-preserving self-homeomorphisms. It records the symmetries of the surface up to the deformations, it acts on the Teichmüller space by biholomorphisms with the moduli space as quotient, it is the orbifold fundamental group of the moduli space of the curves, and it is the receptacle of the monodromy of every family of curves. Its classification theory — the theorem of Nielsen and Thurston that every element is periodic, reducible or pseudo-Anosov — is the algebraic counterpart of the geometry of the surface, and the group is one of the standard examples in geometric group theory: it is finitely presented, its large-scale geometry is governed by its action on the curve complex, and its stable cohomology is the polynomial algebra on the tautological classes by the theorem of Madsen and Weiss.

The article develops the group in the order in which its structure unfolds: the definition, the Dehn twists and the generation and presentation theorems; the classification of the elements, with the periodic, reducible and pseudo-Anosov cases and the Nielsen realisation; the curve complex and the hyperbolicity and acylindricity of the action; the Torelli group and the arithmetic of the homology action; the finiteness and stability properties, with the cohomological dimension, the Torelli and the stable cohomology; and the applications to the moduli spaces, to the fibred three-manifolds and to the outer automorphisms of the free groups. The Teichmüller theory of the companion article is used throughout: the group is studied through its action on the Teichmüller space.

The boundaries of the article. The Teichmüller space, the Teichmüller and Weil–Petersson metrics, the moduli space, the Bers embedding and the compactifications are not developed here; the present article takes the space and its properties as given. The hyperbolic structures, the geodesic laminations, the measured foliations, the convex cores and the Nielsen convex bodies are those of Hyperbolic Geometry. The fibred three-manifolds arising as the mapping tori, the hyperbolic Dehn surgery and the virtual fibring theorem are those; the knots and the braids as topological objects are those; the fixed points of the finite subgroups and the automorphisms of the surfaces are those in the spherical case and of the hyperbolic geometry in the general case. The surface groups, the free groups and the outer automorphism groups are those of Topology on Groups and of the group theory of Part I; the homology and the cohomology of the groups, the algebraic $K$-theory and the group cohomology are those of Algebraic Topology and of the homological algebra of Part I, written by other agents. The measure theory of the measured foliations and the analytic theory of the Thurston boundary belong to Part III, and no physics is invoked.

Definitions, Dehn Twists and Presentations

Definition. Let $S$ be a compact oriented surface. The mapping class group $\mathrm{Mod}(S)$ is the group of the isotopy classes of the orientation-preserving homeomorphisms $S\to S$ that fix the boundary pointwise:

$$ \mathrm{Mod}(S) = \pi_0\,\mathrm{Homeo}^+(S,\partial S) = \mathrm{Homeo}^+(S,\partial S)/\mathrm{Homeo}_0(S,\partial S) . $$

The extended mapping class group $\mathrm{Mod}^{\pm}(S)$ is defined by omitting the orientation condition, and it contains $\mathrm{Mod}(S)$ as a subgroup of index two. For a closed surface the group depends only on the topological type; for a surface with punctures the homeomorphisms are required to fix the punctures setwise, and for a surface with boundary they fix the boundary pointwise. The pure mapping class group fixes the punctures individually.

Definition. Let $a\subseteq S$ be a simple closed curve and let $N(a)$ be a closed annular neighbourhood with a chosen identification with $S^1\times I$. The Dehn twist $T_a$ is the class of the homeomorphism that is the identity outside $N(a)$ and that sends $(e^{i\theta},t)\mapsto(e^{i(\theta+2\pi t)},t)$ on the annulus. It depends only on the isotopy class of $a$, and it is an element of infinite order whose action on the curve complex has positive translation length.

Proposition (the action on homology and the symplectic representation). Let $S$ be closed of genus $g$. The action on $H_1(S;\mathbb{Z}) = \mathbb{Z}^{2g}$ preserves the intersection form and gives a surjection

$$ \mathrm{Mod}(S)\longrightarrow Sp_{2g}(\mathbb{Z}) , $$

whose kernel is the Torelli group $\mathcal{I}(S)$. For $g\geq1$ the map is surjective (for $g=1$ it is an isomorphism $\mathrm{Mod}(T^2)\cong SL_2(\mathbb{Z})$), and the Torelli group is the group of the mapping classes acting trivially on the homology; it is trivial for $g=1$, finitely generated for $g\geq3$ and infinitely generated for $g=2$ (McCullough–Miller).

Proof sketch. A homeomorphism acts on the homology preserving the intersection form; that the resulting map is surjective is proved by exhibiting the images of the Dehn twists, which generate the symplectic group. The properties of the Torelli group — the theorem of Johnson that it is finitely generated for $g\geq3$, and the theorem of McCullough and Miller that it is not finitely generated for $g=2$ — are the subject of the Johnson theory below.

Theorem (Dehn, Lickorish, Humphries; generation by twists). Let $S$ be a closed oriented surface of genus $g\geq1$. Then $\mathrm{Mod}(S)$ is generated by finitely many Dehn twists: the twists about the $3g-1$ curves of a chain generate (Lickorish), and the twists about $2g+1$ explicit curves generate, that number being minimal for $g\geq2$ (Humphries). For the surface with punctures the Dehn twists generate the pure mapping class group, and the full group requires in addition the elements permuting the punctures, which for the disc are the half-twists of the braid groups.

Proof sketch. The key point is that a twist about a curve $a$ followed by the isotopy that pushes $a$ along a curve $b$ with a single intersection expresses the twist about the image curve, so the twists about a connected family of curves generate all the twists; the surface is covered by a chain of the curves of the generating set, and the classification of the elementary moves reduces the general homeomorphism to a product of twists. The minimality of $2g+1$ is the theorem of Humphries.

Theorem (presentations; Wajnryb, Birman–Hilden, Hatcher–Thurston). The mapping class group of a closed oriented surface of genus $g\geq2$ is finitely presented; Wajnryb's presentation has the generators the Dehn twists about the curves of the standard chain and the relations of four explicit types. Equivalently, the group acts on the curve complex with a finite fundamental domain and finite stabilisers generated by the twists, and the presentation is read off from the quotient complex. For the torus the group is $\mathrm{Mod}(T^2)\cong SL_2(\mathbb{Z})$, the amalgamated product $\mathbb{Z}/4*_{\mathbb{Z}/2}\mathbb{Z}/6$ with the presentation $\langle x,y\mid x^4=1,\ x^2=y^3\rangle$; for the disc with $n$ marked points the group is the braid group $B_n$, generated by the half-twists with the braid relations, and the corresponding group for the sphere is the spherical braid group, obtained from $B_n$ by the additional relation of the full twist, with the exceptional cases for small $n$ treated separately.

Proof sketch. The action on the curve complex is cocompact on the $1$-skeleton and the stabilisers of the simplices are virtually generated by the twists; the presentation is obtained from the quotient complex by the Reidemeister–Schreier method. The identification with the braid group is the theorem that a homeomorphism of the disc fixing the boundary pointwise and permuting the marked points is the same datum as a braid, with the full twist generating the centre.

Theorem (the Birman exact sequence). Let $S$ be a surface with a marked point and let $\mathrm{Mod}(S,p)$ be the mapping class group of the punctured surface. Then there is a short exact sequence

$$ 1\longrightarrow \pi_1(S,p)\longrightarrow \mathrm{Mod}(S,p)\longrightarrow \mathrm{Mod}(S)\longrightarrow 1 , $$

the Birman exact sequence, in which the last map forgets the marked point and the first map is the point-pushing map: it realises a loop in $S$ based at $p$ by dragging the surface along the loop, which returns the surface to itself while moving the marked point. Consequently the mapping class group of the punctured surface is an extension of the mapping class group of the closed surface by the surface group itself.

Proof sketch. The long exact sequence of the fibration of the configuration space of one point over the surface gives the sequence; the identification of the kernel with the fundamental group is the point-pushing construction.

The Classification of the Mapping Classes

Theorem (Nielsen–Thurston classification). Let $S$ be a compact oriented surface and let $f\in\mathrm{Mod}(S)$. Then $f$ satisfies exactly one of the following:

(a) $f$ is periodic: it has finite order;

(b) $f$ is reducible: there is a nonempty collection of disjoint essential simple closed curves, no two isotopic and none parallel to the boundary, that $f$ permutes, and $f$ is not periodic;

(c) $f$ is pseudo-Anosov: there is a pair of transverse measured foliations on $S$ such that $f$ preserves the pair and multiplies the transverse measures by $\lambda$ and $\lambda^{-1}$ for some $\lambda>1$, the stretch factor.

The classification is invariant under conjugation, and the three classes are distinguished by the growth of the word length and by the translation length in the curve complex: the periodic elements have zero growth, the reducible elements have linear growth confined to the blocks of the reduction, and the pseudo-Anosov elements grow exponentially. Consequently $\mathrm{Mod}(S)$ is not hyperbolic for $g\geq2$: two Dehn twists about disjoint curves generate $\mathbb{Z}^2$, which cannot be quasi-isometrically embedded in a hyperbolic space.

Proof sketch. The proof is by the action of $f$ on the space of the projective measured foliations, the Thurston boundary of the Teichmüller space: a comparison of the lengths of the iterates shows that the action has a fixed point, which is either an invariant curve system (the reducible case), a pair of transverse foliations (the pseudo-Anosov case) or a finite orbit (the periodic case). The alternative is the dynamical dichotomy of the action on the boundary, and the measure theory of the foliations belongs to Part III.

Theorem (Nielsen realisation; Kerckhoff, Thurston). Every periodic mapping class is realised by an isometry: given a finite subgroup $G\subseteq\mathrm{Mod}(S)$ there is a marked hyperbolic structure on $S$ invariant under the action, so that the group acts by isometries and has a fixed point in the Teichmüller space. The maximal order of a cyclic subgroup of $\mathrm{Mod}(S_g)$ for $g\geq2$ is $4g+2$ (Wiman), and the maximal order of a finite subgroup is $84(g-1)$ (Hurwitz); for $g=1$ the orders are those of the elliptic elements of $SL_2(\mathbb{Z})$, namely $1,2,3,4,6$.

Proof sketch. The realisation problem reduces to finding a fixed point for the action of the finite group on the Teichmüller space; the solution constructs the fixed point as the minimiser of the sum of the Teichmüller distances to the images of a base point, an argument of Kerckhoff using the convexity of the length functions along the Teichmüller geodesics. The order bounds are the Gauss–Bonnet computations of the orbifolds $X/G$.

Theorem (Thurston; the mapping torus of a pseudo-Anosov). Let $f\in\mathrm{Mod}(S)$ be pseudo-Anosov with stretch factor $\lambda$. Then the mapping torus

$$ M_f = S\times[0,1]\big/(x,1)\sim(f(x),0) $$

is a closed three-manifold that is hyperbolic, the mapping class $f$ is the monodromy of the fibration and the stretch factor is the exponential growth rate of the lengths of the curves in the fibres; the topological entropy of the monodromy is $\log\lambda$. The mapping torus of a periodic element is a Seifert fibred manifold and that of a reducible element decomposes along the invariant curves, so the trichotomy of the classification is the trichotomy of the geometry of the fibred three-manifolds.

Proof sketch. The suspension of the pseudo-Anosov homeomorphism has a flow that is hyperbolic on a subset and the manifold admits a hyperbolic structure with the fibres as the immersed totally geodesic surfaces; the construction of the metric is the double limit theorem of Thurston. The geometry of the hyperbolic three-manifolds is that of Hyperbolic Geometry and the classification of the fibred manifolds is deferred to Part III.

The Curve Complex and the Geometry of the Group

Definition. The curve complex $\mathcal{C}(S)$ is the simplicial complex whose vertices are the isotopy classes of the essential simple closed curves and whose simplices are the sets of pairwise disjoint curves; it is $(3g-4+n)$-dimensional for a surface of genus $g$ with $n$ punctures in the non-exceptional cases, and the mapping class group acts on it by automorphisms. The action on the vertices extends to the natural action on the Teichmüller space: the length function $\ell_\gamma$ is a function on $\mathcal{T}(S)$, and the geometry of the curve complex records the coarse geometry of the lengths.

Theorem (Masur–Minsky; Ivanov; Bowditch). Let $S$ have genus $g\geq2$ with $n$ punctures. Then:

(a) the curve complex $\mathcal{C}(S)$ is $\delta$-hyperbolic in the sense of Gromov (Masur–Minsky), and its Gromov boundary is the space of the minimal filling laminations;

(b) the action of $\mathrm{Mod}(S)$ on $\mathcal{C}(S)$ is acylindrical (Bowditch), so that the group admits a non-elementary acylindrically hyperbolic action;

(c) every simplicial automorphism of $\mathcal{C}(S)$ is induced by a mapping class (Ivanov), so that $\mathrm{Mod}(S)$ is the full automorphism group of the complex in the non-exceptional cases.

Proof sketch. The hyperbolicity is proved by the construction of the hyperbolic geodesics in the complex from the Teichmüller geodesics and the distance formula relating the distance in the complex to the lengths of the curves; the acylindricity is the boundedness of the intersection of the pointwise stabilisers of far-apart pairs of vertices. The rigidity statement of Ivanov is proved by the reconstruction of the surface from the complex.

Theorem (Masur–Minsky; the distance formula and the rank). The distance in the Teichmüller metric of two points with large distance in the curve complex is comparable to the sum of the distances of their projections to the curve complexes of the subsurfaces; this distance formula governs the coarse geometry of the mapping class group. Consequently the group is not hyperbolic but is hierarchically hyperbolic (Behrstock–Hagen–Sisto): it has a structure of a space with a family of hyperbolic spaces and projection maps satisfying the axioms of the theory, and the quasi-flats in the group are confined to the stabilisers of the subsurfaces, so that the quasi-isometric rigidity of the group reduces to the rigidity of the subgroups.

Proof sketch. The distance formula is proved by the construction of the hierarchy of the geodesics in the curve complexes of the subsurfaces and the verification that the conjectured distance is comparable to the Teichmüller distance; the hierarchical structure is obtained from the family of the curve complexes of the subsurfaces and the projection maps. The analytic input of the proof is the comparison between the Teichmüller distance and the curve-complex distance, which is stated as standard.

Remark (the large-scale geometry). The mapping class group is not hyperbolic, not amenable for $g\geq2$, has exponential growth, and its stable commutator length and its bounded cohomology are the subject of the theory of the group; the Torelli group — the kernel of the symplectic representation, hence normal — is not nilpotent, the Johnson filtration measuring the failure of the nilpotency; the word metric, the translation length and the growth rates are the invariants of the geometric group theory of the surface. The action on the curve complex gives the group a non-elementary acylindrical hyperbolic action, and the classification theorem of the elements of the group is the statement that the translation length in the curve complex vanishes exactly for the periodic and the reducible elements.

The Torelli Group, Cohomology and Stability

Theorem (Johnson; the Torelli group). Let $\mathcal{I}(S)$ be the Torelli group, the kernel of the symplectic representation of $\mathrm{Mod}(S)$ for a closed surface of genus $g$. Then $\mathcal{I}(S)$ is trivial for $g=1$, is not finitely generated for $g=2$, and is finitely generated for $g\geq3$, generated by the twists about the separating curves and the BP-maps (the "bounding pair" maps); its abelianisation for $g\geq3$ is computed by the Johnson homomorphism

$$ \tau : \mathcal{I}(S)/[\mathcal{I}(S),\mathcal{I}(S)] \longrightarrow \textstyle\bigwedge^3 H_1(S;\mathbb{Z}) , $$

which is an isomorphism onto the kernel of the contraction in the exterior algebra, and the Johnson filtration of the group by the kernels of the actions on the successive nilpotent quotients of the surface group is the measure of the failure of the group to be nilpotent.

Proof sketch. The Torelli group acts on the homology trivially, so it acts on the second nilpotent quotient of the surface group; the Johnson homomorphism is the resulting map to the exterior algebra, and its surjectivity and injectivity on the abelianisation are the computations of Johnson. The higher terms of the filtration are the subject of the Johnson theory and of the conjecture on the lower central series of the Torelli group.

Theorem (Harer; the cohomological dimension and stability). Let $S$ be a closed surface of genus $g\geq2$. Then $\mathrm{Mod}(S)$ is finitely presented and has virtual cohomological dimension $4g-5$; for a surface with $n\geq1$ punctures the virtual cohomological dimension is $4g-4+n$. The homology $H_i(\mathrm{Mod}(S_g);\mathbb{Z})$ is independent of $g$ for $g$ large with $i$ fixed (Harer stability), so that the stable cohomology is the cohomology of the limiting object. The first homology is

$$ H_1(\mathrm{Mod}(S_g);\mathbb{Z}) = \begin{cases} 0 & g\geq3 ,\\ \mathbb{Z}/10 & g=2 ,\\ \mathbb{Z}/12 & g=1 ,\end{cases} $$

the genus-one case being the abelianisation of $SL_2(\mathbb{Z})$, and the mapping class group of a closed genus-$g$ surface with $g\geq3$ has trivial abelianisation (Powell).

Proof sketch. The finiteness of the presentation is Wajnryb's theorem; the virtual cohomological dimension is the dimension of the Teichmüller space quotient by the stabilisers, which are finite; the stability is proved by the comparison of the complexes of the curves of the surfaces of successive genera, with the maps induced by the stabilisation of the surface; the first homology is computed from the abelianisation, which is generated by the twists subject to the relations of Powell and of Mumford.

Theorem (Madsen–Weiss; the Mumford conjecture). The stable rational cohomology of the mapping class groups is a polynomial algebra on the Miller–Morita–Mumford classes $\kappa_1,\kappa_2,\ldots$,

$$ H^*(\mathrm{Mod}(S_g);\mathbb{Q}) = \mathbb{Q}[\kappa_1,\kappa_2,\ldots], \qquad \deg\kappa_i = 2i , $$

in the stable range; equivalently, the classifying space of the stable mapping class group is the infinite loop space of the spectrum $\mathbb{CP}^\infty_{-1}$. Consequently the tautological classes generate the stable cohomology, and the proof of the Mumford conjecture is the theorem of Madsen and Weiss via the homotopy theory of the surfaces and the Segal conjecture.

Proof sketch. The Mumford conjecture is translated into a computation of the homotopy type of the cobordism category of the surfaces, and the stable homotopy of the category is identified with the spectrum; the calculation of the cohomology of the mapping class group follows from the homotopy-theoretic computation. The homotopy theory belongs to Algebraic Topology, and the Grothendieck–Teichmüller group is the related automorphism group of the tower of the mapping class groups.

Examples and Applications

Example (the torus and the modular group). For the torus, $\mathrm{Mod}(T^2) = SL_2(\mathbb{Z})$, generated by the two Dehn twists about the meridian and the longitude with the relations of the amalgamated product above; the elements are elliptic (finite order $1,2,3,4,6$), parabolic (the conjugates of the Dehn twists) or hyperbolic (the Anosov classes), in agreement with the Nielsen–Thurston trichotomy, and the quotient of the Teichmüller space is the modular curve. The mapping torus of a hyperbolic element is a Sol manifold, of a parabolic element a Nil or a Seifert fibred manifold, and of an elliptic element a Seifert fibred manifold with the spherical or the Euclidean geometry, which is the three-dimensional case of the classification of the mapping tori.

Example (the sphere with punctures and the braid groups). For the disc with $n$ marked points the mapping class group is the braid group $B_n$, generated by the half-twists $\sigma_1,\ldots,\sigma_{n-1}$ with the braid relations; the pure braid group is the pure mapping class group; and the point-pushing sequence exhibits the braid groups as the extensions of the punctured mapping class groups by the free groups. The Birman–Hilden theorem identifies the hyperelliptic mapping class group with the lift of the braid group, and the two theories are thus the same theory of the configurations of the points and their deformations.

Example (the hyperelliptic involution and the genus two case). Every closed surface of genus two is hyperelliptic, and the hyperelliptic involution generates the centre of $\mathrm{Mod}(S_2)$; the quotient of the surface by the involution is the sphere with six branch points, and by the Birman–Hilden theory the hyperelliptic mapping class group is the lift of the mapping class group of the six-punctured sphere through the covering, so that $\mathrm{Mod}(S_2)$ is described as a central extension of the corresponding group of the sphere by the involution. The example is the smallest in which the Torelli group is infinitely generated and the smallest in which the level-$2$ subgroup is related to the arithmetic group of the orthogonal type.

Application (monodromy and fibred three-manifolds). A fibred three-manifold with fibre $S$ has a monodromy in $\mathrm{Mod}(S)$, well defined up to conjugacy; the geometry of the manifold is determined by the Nielsen–Thurston class of the monodromy, and the virtual fibring theorem states that every closed hyperbolic three-manifold is virtually fibred, so its finite covers have monodromies in a mapping class group. The action of the monodromy on the homology of the fibre computes the homology of the manifold and the Alexander polynomial of the knot in the case of the complement of a fibred knot.

Application (the outer automorphism group of the free group). The Dehn–Nielsen–Baer theorem realises $\mathrm{Mod}(S_{g,1})$ as a subgroup of $\mathrm{Out}(F_{2g})$; the image is the subgroup of the outer automorphisms preserving the conjugacy class of the boundary curve, equivalently acting by a symplectic map on the abelianisation, and the identification of the two groups, together with the classification of the elements of $\mathrm{Out}(F_n)$ in the style of the Nielsen–Thurston theory, is one of the bridges between the surface topology and the combinatorial group theory of Topology on Groups. The Torelli group is the analogue of the IA-automorphism group, and the Johnson homomorphism is the analogue of the Andreadakis–Johnson filtration of $\mathrm{Aut}(F_n)$.

Summary

The mapping class group $\mathrm{Mod}(S)$ is the group of the isotopy classes of the orientation-preserving homeomorphisms of the surface; it is generated by the Dehn twists (Dehn, Lickorish, Humphries, with the minimal number $2g+1$ for a closed surface), it is finitely presented (Wajnryb), and its action on the homology gives the symplectic representation with the Torelli group as kernel. The classification theorem of Nielsen and Thurston splits every mapping class into the periodic, the reducible and the pseudo-Anosov classes; the periodic ones are realised by the isometries of a hyperbolic structure (Nielsen realisation, Kerckhoff), with the maximal cyclic order $4g+2$ and the maximal finite group order $84(g-1)$; the pseudo-Anosov ones are the exponential-growth classes, with a stretch factor $\lambda>1$, and their mapping tori are the hyperbolic fibred three-manifolds (Thurston). The classification is the dynamical counterpart of the trichotomy of the three-dimensional geometries of the mapping tori.

The large-scale geometry of the group is governed by the curve complex: it is $\delta$-hyperbolic by Masur and Minsky, the action is acylindrical by Bowditch, the automorphism group is the group itself by Ivanov, and the group is hierarchically hyperbolic with the distance formula of Masur and Minsky controlling the coarse geometry. The Torelli group is generated by the twists about the separating curves and the BP-maps, with the abelianisation computed by the Johnson homomorphism into the exterior algebra; the group has virtual cohomological dimension $4g-5$, its homology satisfies the stability theorem of Harer in the degree fixed as the genus grows, its first homology is $\mathbb{Z}/10$ for genus two and $0$ for $g\geq3$, and its stable rational cohomology is the polynomial algebra on the Miller–Morita–Mumford classes by the theorem of Madsen and Weiss. The applications run from the moduli spaces of the marked hyperbolic structures through the fibred manifolds to the outer automorphisms of the free groups.

Summary of Notation

Symbol	Meaning
$S$, $S_g$, $S_{g,n}$	Oriented surface of genus $g$ with $n$ punctures or boundary components
$\mathrm{Mod}(S)$, $\mathrm{Mod}^{\pm}(S)$	Mapping class group; extended mapping class group
$T_a$	Dehn twist about the curve $a$
$\mathrm{Homeo}_0(S,\partial S)$	Homeomorphisms isotopic to the identity relative to the boundary
$Sp_{2g}(\mathbb{Z})$, $H_1(S;\mathbb{Z})$	Symplectic group; homology of the surface with the intersection form
$\mathcal{I}(S)$, $\tau$	Torelli group; Johnson homomorphism to $\bigwedge^3 H_1$
$\mathcal{C}(S)$	Curve complex; vertices the essential simple closed curves, simplices the disjoint families
$\lambda$	Stretch factor of a pseudo-Anosov class; topological entropy $\log\lambda$
$M_f$	Mapping torus of $f$; hyperbolic iff $f$ is pseudo-Anosov
$4g-5$, $4g-4+n$	Virtual cohomological dimension of $\mathrm{Mod}(S_g)$ and $\mathrm{Mod}(S_{g,n})$, $n\geq1$
$4g+2$, $84(g-1)$	Maximal cyclic order (Wiman); maximal finite group order (Hurwitz)
$\kappa_i$	Miller–Morita–Mumford classes; generate the stable cohomology (Madsen–Weiss)
$B_n$, $\sigma_i$	Braid group and its generators; $\mathrm{Mod}(D^2,n)=B_n$; the spherical braid group for the sphere