BRAIDING IN GROUPS AND MAPPING CLASS GROUPS

Vladimir V. Vershinin

Abstract. The series of the mapping class groups Γ_{g,1} of a surface of the genus g with 1 component of the boundary gives a natural example of a braiding in groups, as defined by Ross Street.

Ross Street introduced recently an interesting and natural braiding in the category of groups [St]. The purpose of this note is to show that this kind of braiding naturally arises in the mapping class groups. This language allows clearly express the structure firstly considered by E. Miller [Mil] in the mapping class groups. We remind briefly the R. Street’s construction and results.

Let Gp_{∗} be the category of pointed groups. Their objects are pairs (G, g_{0}) where G is a group and g_{0} is its element. A morphism

α : (G, g_{0}) → (H, h_{0})

is a group homomorphism α : G → H, such that α(g_{0}) = h_{0}. The usual monoidal category of groups Gp is identified with the full subcategory of Gp_{∗} consisting of the objects (G, g_{0}) with g_{0} is equal to the unit of the groups G. The monoidal (tensor) structure on Gp_{∗} is given by the free product on groups and group product on elements:

(G, g_{0}) ⊗ (H, h_{0}) = (G ∗ H, g_{0}h_{0}).

Theorem (R. Street [St]). The monoidal category Gp_{∗} admits a braiding

c_{G,H} : (G ∗ H, g_{0}h_{0}) → (H ∗ G, h_{0}g_{0})

which takes g ∈ G to h_{0}gh_{0}^{−1} and h ∈ H to h ∈ B ∗ H.

It is possible to consider monoids pointed by invertible elements instead of pointed groups. Then the larger category is obtained.

Mapping class groups are studied widely in present because of their relations with various fields of mathematics: topology, complex analysis, algebraic geometry, group theory. Alternatively these groups are known as the Teichmüller modular groups [I], and also as the homeotopy group of a surface. The general case is the mapping class group Γ_{g,k}^{n} of a surface S_{g} of the genus g with k components of the boundary and n fixed (as a set) points. Relations between mapping class groups and

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 20F38, 20F36, 18D10.

Key words and phrases. Braid group, mapping class group, free group.

The research of was supported in part by RFFR Grant 96-01-016331 and INTAS

braid groups were studied also intensively (see the survey [Bi4]). We mention the following of them. At first the group Γ_{0,1}^{n} is the classical braid group on n strings. Then any closed surface of the genus g can be obtained as a branched covering space of the sphere with 2g+2 branching points and braid homeomorphisms from the base space (corresponding to generators of Γ_{0,1}^{2g+2}) can be lifted to homeomorphisms of surfaces and generate subgroups of the mapping class group of the surface [BH1], [BH2], [Vi]. Thirdly there are connections between the braid groups of surfaces Br_{m}(S_{g}), where the number of strings m is equal either to the number of points n or to the number of connected components of the boundary k. This situation was studied by Joan Birman [Bi1, Bi3] and G. P. Scott [Sc]. Other relations are based on presentations of mapping class groups [Bi2]. The standard generators of the mapping class group which correspond to the Dehn twists along the certain curves on a surface satisfy braid relations (among the others) [W]. This gives possibility to construct homomorphisms from braid groups to mapping class groups. A lot of information about the mapping class groups is contained also in the surveys [Co], [Mis].

The mapping class group Γ_{g,k}^{n} of a surface of the genus g with k components of the boundary and n fixed points is defined geometrically as follows. Let Diff_{+}(S_{g}; k; n) be the group of orientation preserving diffeomorphisms of a surface S_{g} of the genus g which fix pointwise the k components of the boundary and have n points fixed as a set. We denote by Diff_{+}^{0}(S_{g}; k; n) the connected component of the identity diffeomorphism. Then the mapping class group Γ_{g,k}^{n} is defined as the factor group

Diff_{+}(S_{g}; k; n)/Diff_{+}^{0}(S_{g}; k; n) = π_{0}(Diff_{+}(S_{g}; k; n)).

There are equivalent definitions of the mapping class groups in terms of homeomorphisms and homotopy equivalences. Let Homeo(S_{g}; k; n) be the group of homeomorphisms of a surface S_{g} of the genus g which fix pointwise the k components of the boundary and have n points fixed as a set. A map f ∈ Homeo(S_{g}; k; n) is said to be orientation preserving, if the induced homomorphism in two-dimensional homology

H(f) : H_{2}(S_{g}, ∂S_{g}; ℤ) → H_{2}(S_{g}, ∂S_{g}; ℤ)

is the identity map of the group H_{2}(S_{g}, ∂S_{g}; ℤ) ≅ ℤ. We denote by Homeo_{+}(S_{g}; k; n) the subgroup of orientation preserving homeomorphisms from Homeo(S_{g}; k; n) and by Homeo_{+}^{0}(S_{g}; k; n) the connected component of the identity homeomorphism. Then the mapping class group Γ_{g,k}^{n} is isomorphic to the factor group

Homeo_{+}(S_{g}; k; n)/Homeo_{+}^{0}(S_{g}; k; n) = π_{0}(Homeo_{+}(S_{g}; k; n)).

We omit writing zero in the notation Γ_{g,k}^{n}, if n or k is equal to zero. The group Γ_{g} acts on the Teichmüller space T^{3g−3} properly and discontinuously. The quotient of this action is the moduli space of smooth algebraic curves of the genus g.

Let F (S_{g}, n) be the space of n-tuples of pairwise different points in S_{g}:

F (S_{g}, n) = {p_{1}, ..., p_{n}) ∈ S_{g}^{n} : p_{i} ≠ p_{j} for i ≠ j}.

If we fix n different points on a surface S_{g} then the evaluation map

Homeo_{+}(S_{g}; 0; 0) → F (S_{g}, n)

is defined. It was proved by Joan Birman [Bi1, Bi3], that this map is a locally trivial fibering with the fiber equal to Homeo_{+}(S_{g}; 0; n). The exactness of the following sequence is a corollary of this fact:

(1)  ... → π_{1}(F (S_{g}, n)) → Γ_{g}^{n} → Γ_{g} → e.

In the case when n = 1, g ≥ 2, the following short exact sequence

e → π_{1}(S_{g}) → Γ_{g}^{1} → Γ_{g} → e

is obtained from (1). We have also the exact sequence

(2)  e → ℤ → Γ_{g,1} → Γ_{g}^{1} → e,

where the generator of ℤ corresponds to the Dehn twist along the boundary of the disk. Using the construction of “pair of pants” one can obtain an inclusion

(3)  Γ_{g,1} → Γ_{g+1,1}

and a pairing

(4)  μ_{g,q} : Γ_{g,1} × Γ_{q,1} → Γ_{g+q,1}.

Using the inclusions (3) one defines the stable mapping class group

Γ_{∞,1} = ∪_{g}Γ_{g,1}.

The mapping class group Γ_{g}^{n} of a surface of the genus g with n points deleted may be interpreted algebraically as follows. Let Π(g, n) be the group with generators a_{1}, b_{1}, ..., a_{g}, b_{g}, u_{1}, ..., u_{n} and one defining relation

u_{1}u_{2}...u_{n}a_{1}b_{1}a_{1}^{−1}b_{1}^{−1}...a_{g}b_{g}a_{g}^{−1}b_{g}^{−1} = 1.

We denote by A(g, n) the group of those automorphisms of the group Π(g, n) which map each generator u_{i} into its conjugate or to the conjugate of another such generator u_{j}. Let I(g, n) be the group of inner automorphisms of the group Π(g, n). Then I(g, n) is the normal subgroup of A(g, n) and there is an isomorphism (see, for example, [MKS, p. 175]):

(5)  Γ_{g}^{n} ≅ A(g, n)/I(g, n).

Suppose now that n = 1. Then we interpret the group A(g, n) as the group of automorphisms of the free group F_{2g} with generators a_{1}, b_{1}, ..., a_{g}, b_{g}, such that the elements of A(g, n) map the commutator a_{1}b_{1}a_{1}^{−1}b_{1}^{−1}...a_{g}b_{g}a_{g}^{−1}b_{g}^{−1} into its conjugate. Then from the formulas (2) and (5) one can identify the group Γ_{g,1} with the group of automorphisms ϕ of the free group F_{2g} for which the commutator is invariant

(a_{1}b_{1}a_{1}^{−1}b_{1}^{−1}...a_{g}b_{g}a_{g}^{−1}b_{g}^{−1})ϕ = a_{1}b_{1}a_{1}^{−1}b_{1}^{−1}...a_{g}b_{g}a_{g}^{−1}b_{g}^{−1}.

The generator of the group ℤ by the map from (2) corresponds to an element in A(g, n), which is conjugation by the commutator a_{1}b_{1}a_{1}^{−1}b_{1}^{−1}...a_{g}b_{g}a_{g}^{−1}b_{g}^{−1}.

The canonical imbeddings

Aut F_{2g} × Aut F_{2q} → Aut F_{2(g+q)}

induce the pairing for the system of groups Γ_{g,1}:

μ_{g,q} : Γ_{g,1} × Γ_{q,1} → Γ_{g+q,1},

in such a way that the following diagram

Γ_{g,1} × Γ_{q,1}  ⟶^{μ_{g,q,}} Γ_{g+q,1}
↓  ↓
Aut F_{2g} × Aut F_{2q} ⟶ Aut F_{2(g+q)}

is commutative. This pairing is associative in a usual way:

Γ_{g,1} × Γ_{q,1} × Γ_{l,1} ⟶^{Id×μ_{q,l}} Γ_{g,1} × Γ_{q+l,1}
↓ μ_{g,q} × Id  ↓ μ_{g,q+l}
Γ_{g+q,1} × Γ_{l,1}  ⟶^{μ_{g+q,l}} Γ_{g+q+l}.

It coincides with the pairing (4).

These data define in a standard way a strict monoidal category GM whose objects are free groups F_{2g}, g = 0, 1..., and morphisms are defined by the formula:

hom_{GM}(F_{2k}, F_{2l}) = \{ \begin{matrix}Γ_{k,1}, & if k = l, \\ ∅, & if k ≠ l. \end{matrix}

We suppose that Γ_{0,1} = {e}.

Our considerations can be summarized by the following

Proposition 1. The monoidal category GM naturally imbeds into the monoidal category of pointed groups Gp_{∗} by the functor which maps F_{2k} to the pair

(F_{2k}, a_{1}b_{1}a_{1}^{−1}b_{1}^{−1}...a_{k}b_{k}a_{k}^{−1}b_{k}^{−1})

and imbeds Γ_{k,1} into the subgroup of automorphisms ϕ of the free group F_{2k} for which the commutator is invariant

(a_{1}b_{1}a_{1}^{−1}b_{1}^{−1}...a_{k}b_{k}a_{k}^{−1}b_{k}^{−1})ϕ = a_{1}b_{1}a_{1}^{−1}b_{1}^{−1}...a_{k}b_{k}a_{k}^{−1}b_{k}^{−1}.

The braiding in Gp_{∗} defines a braiding in GM, so that the later becomes a braided monoidal category.

The classical representation

ρ_{g} : Γ_{g,1} → Sp_{2g}(ℤ)

geometrically is defined by the action of a homeomorphism or homotopy equivalence on the first homology group of the underlying surface. In the interpretation that we use the representation ρ_{g} is obtained from the canonical epimorphism

F_{2g} → A_{2g},

where A_{2g} is a free abelian group of the rank 2g. We denote by Sp the analogous to GM category defined with the help of the modular groups. Representations ρ_{g} define a morphism of monoidal categories:

P : GM → Sp.

It follows from [JS] that the braid category B is the free braided monoidal category generated by the single object ¯1. In the category B objects {¯0, ..., k̄, ...} correspond to integer numbers from zero to infinity and morphisms are defined by the formula:

hom_{B}(k̄, l̄) = \{ \begin{matrix}Br_{k}, & if k = l, \\ ∅, & if k ≠ l, \end{matrix}

where Br_{k} is the braid group on k strings. The monoidal structure is defined by addition of numbers and addition of braids.

Let us consider the free group F_{2} on the generators a and b as an object of Gp_{∗} by choosing the commutator aba^{−1}b^{−1} to be the distinguished point. The freeness of B allows to define a strong monoidal functor R : B → Gp_{∗} unique up to isomorphism with R(¯1) = F_{2}. It follows that there is a cannonical isomorphism R(k̄) ≅ F_{2k}, where the distinguished point in F_{2k} is the product a_{1}b_{1}a_{1}^{−1}b_{1}^{−1}...a_{k}b_{k}a_{k}^{−1}b_{k}^{−1}.

Proposition 2. The functor

R : B → Gp_{∗}

is faithful.

The proof is the same as in [V].

Remark. These consideration are completely analogous to those from the Proposition 2 in [St].

The category Sp can also be considered as a subcategory of Gp_{∗} where the fixed elements of the groups A_{2k} are the units. Then the functor P : GM → Sp respects the distinguished elements in groups.

If the fixed element g_{0} ∈ G is equal to the unit, then the symmetry as in the Theorem of R. Street defines a permutative category structure on Gp. Let us consider its subcategory A consisting of the free groups F_{n} as objects (one for each n) and morphisms defined by the formula:

hom_{A}(F_{k}, F_{l}) = \{ \begin{matrix}Aut F_{k}, & if k = l, \\ ∅, & if k ≠ l. \end{matrix}

Then it will be a symmetric category and analogously to the Proposition 2 there exists a functor from the category Σ defined by symmetric groups to A:

Σ → A

which is faithful. This expresses the fact that there is a canonical inclusion of the symmetric group in the group of automorphisms of the free group.

The author is thankful to R. Street and M. Batanin for the information and discussions.

References

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[W] B. Wajnryb, A simple presentation for the mapping class groups of an orientable surface, Israel J. of Math. 45 (1983), 157–174.

Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

   E-mail address: versh@math.nsc.ru