THE BRAID STRUCTURE OF MAPPING CLASS GROUPS

Zbigniew Fiedorowicz and Yongjin Song

0. Introduction

It was shown by Stasheff([13]) and MacLane([7]) that monoidal categories give rise to loop spaces. A recognition principle specifies an internal structure such that a space X has such a structure if and only if X is of the weak homotopy type of n-fold loop space. It has been known for years that there is a relation between coherence problems in homotopy theory and in categories. May’s recognition theorem([9]) states that for little n-cube operad C_{n}, n ≥ 2, every n-fold loop space is a C_{n}-space and every connected C_{n}-space has the weak homotopy type of an n-fold loop space.

E. Miller([9]) observed that there is an action of the little square operad on the disjoint union of BDiff^{+}(S_{g,1})’s extending the F -product which is induced by a kind of connected sum of surfaces. We hence have that the group completion of ⨿_{g≥0}BDiff^{+}(S_{g,1}) is a double loop space up to homotopy. Miller applied this result to the calculation of the homology groups of mapping class groups. However his description of the action of the little square operad is somewhat obscure. On the other hand the first author proved([4]) that the group completion of the nerve of a braided monoidal category is the homotopy type of a double loop space. This result implies that there exists a strong connection between braided monoidal category and the mapping class groups Γ_{g,1} in view of Miller’s result.

We, in this paper, show that the disjoint union of Γ_{g,1}’s is a braided monoidal category with the product induced by the connected sum. Hence the group completion of ⨿_{g≥0}BΓ_{g,1} is the homotopy type of a double loop space. We explicitly describe the braid structure of ⨿_{g≥0}Γ_{g,1}, regarding Γ_{g,1} as the subgroup of the automorphism group of π_{1}S_{g,1} that consists of the automorphisms fixing the fundamental relator. We provide the formula for the braiding (Lemma 2.1) which is useful in dealing with the related problems. Using this braiding formula (2.2), we can make a correction on Cohen’s diagram. We also show that the double loop space structure of the disjoint union of classifying spaces of mapping class groups cannot be extended to the triple loop space structure (Theorem 2.5). It seems important to note the relation between the braid structure and the double loop space structure in an explicit way.

Turaev and Reshetikhin introduced an invariant of ribbon graphs which is derived from the theory of quantum groups and is a generalization of Jones polynomial. This invariant was extended to those of 3-manifolds and of mapping class groups(cf.[11],[12],[6]). The definitions are abstract and a little complicated since they are defined through quantum groups. G.

The second author was partially supported by GARC-KOSEF.

22 ZBIGNIEW FIEDOROWICZ AND YONGJIN SONG

Wright([16]) computed the Reshetikhin-Turaev invariant of mapping class group explicitly in the case r = 4, that is, at the sixteenth root of unity. For each h ∈ Γ_{g,0} we can find the corresponding (colored) ribbon graph, whose Reshetikhin-Turaev invariant turns out to be an automorphism of the 1-dimensional summand of V^{k_{1}}⊗V^{k_{1}^{∗}}⊗ ⋯ ⊗V^{k_{g}}⊗V^{k_{g}^{∗}} which we denote by V_{r,g}. We get this ribbon graph using the Heegaard splitting and the surgery theory of 3-manifolds. Wright showed as a result of her calculation that the restriction of this invariant to the Torelli subgroup of Γ_{g,0} is equal to the sum of the Birman-Craggs homomorphisms. dim(V_{4,g}) = 2^{g−1}(2^{g} +1), so the Reshetikhin-Turaev invariant of h ∈ Γ_{g,0}, when r = 4, is a 2^{g−1}(2^{g} +1)×2^{g−1}(2^{g} +1) matrix with entries of complex numbers. Wright proved a very interesting lemma that there is a natural one-to-one correspondence between the basis vectors of V_{4,g} and the ℤ/2-quadratic forms of Arf invariant zero. It would be interesting to check if the Reshetikhin-Turaev representation preserves the braid structure.

1. Mapping class groups and monoidal structure

Let S_{g,k} be an orientable surface of genus g obtained from a closed surface by removing k open disks. The mapping class group Γ_{g,k} is the group of isotopy classes of orientation preserving self-diffeomorphisms of S_{g,k} fixing the boundary of S_{g,k} that consists of k disjoint circles. Let Diff^{+}(S_{g,k}) be the group of orientation preserving self-diffeomorphisms of S_{g,k}. We also have the following alternative definition :

Γ_{g,k} = π_{o}Diff^{+}(S_{g,k})

We will mainly deal with the case k = 1 and k = 0. Γ_{g,1} and Γ_{g,0} are generated by 2g+1 Dehn twists(cf.[14]). There is a surjective map Γ_{g,1} → Γ_{g,0}.

Figure 1. Dehn twists

Many topologists are interested in the homology of mapping class groups. An interesting observation is that there is a product on the disjoint union of Diff^{+}(S_{g,1})’s. It is known by Stasheff([13]) and MacLane([7]) that if a category C has a monoidal structure then its classifying space gives rise to a space which has the homotopy type of a loop space. Fiedorowicz showed([4]) that a braid structure gives rise to a double loop space structure. We now recall the definition of (strict) braided monoidal category.

Definition 1.1. A (strict) monoidal (or tensor) category (C, ⊗, E) is a category C together with a functor ⊗ : C × C → C (called the product) and an object E (called the unit object) satisfying

(a) ⊗ is strictly associative

(b) E is a strict 2-sided unit for ⊗

Definition 1.2. A monoidal category (C, ⊗, E) is called a (strict) braided monoidal category if there exists a natural commutativity isomorphism C_{A,B}: A⊗B → B⊗A satisfying

(c) C_{A,E} = C_{E,A} = 1_{A}

(d) The following diagrams commute:

A⊗B⊗C  ⟶^{C_{A⊗B,C}}  C⊗A⊗B
1_{A}⊗C_{B,C} ↘  ↗ _{C_{A,C}⊗1_{B}}
A⊗C⊗B
A⊗B⊗C  ⟶^{C_{A,B⊗C}}  B⊗C⊗A
C_{A,B}⊗1_{C} ↘  ↗ _{1_{B}⊗C_{A,C}}
B⊗A⊗C

The first author recently gave a proof of the following lemma([4]).

Lemma 1.3. The group completion of the nerve of a braided monoidal category is the homotopy type of a double loop space. The converse is true.

Miller claimed in [10] that there is an action of the little square operad of disjoint squares in D^{2} on the disjoint union of the BΓ_{g,1}’s extending the F -product that is induced by the connected sum. Here the F -product Γ_{g,1} ×Γ_{h,1} ⟶ Γ_{g+h,1} is obtained by attaching a pair of pants (a surfaces obtained from a sphere by removing three open disks) to the surfaces S_{g,1} and S_{h,1} along the fixed boundary circles and extending the identity map on the boundary to the whole pants. Hence, according to May’s recognition theorem on the loop spaces([9]), the group completion of ⨿_{g≥0}BΓ_{g,1} is homotopy equivalent to a double loop space. Miller’s proposition seems correct, although the details are not so transparent. In view of lemma 1.3, the disjoint union of Γ_{g,1}’s should be related to a braided monoidal category. Here we regard ⨿_{g≥0}Γ_{g,1} as a category whose objects are [g], g ∈ ℤ_{+} and morphisms satisfy

Hom([g], [h]) = \{ \begin{matrix}Γ_{g,1} & if g = h \\ ∅ & if g ≠ h \end{matrix}

Without speaking of the action of little square operad, we are going to show that the group completion of ⨿_{g≥0}BΓ_{g,1} is homotopy equivalent to a double loop space by showing that the disjoint union of Γ_{g,1}’s is a braiding monoidal category.

Lemma 1.4. The disjoint union of Γ_{g,1}’s is a braided monoidal category with the product induced by the F -product.

Proof. Let x_{1}, y_{1}, …, x_{g}, y_{g} be generators of the fundamental group of S_{g,1} which are induced by the Dehn twists a_{1}, b_{1}, ⋯, a_{g}, b_{g}, respectively. The mapping class group Γ_{g,1} can be identified with the subgroup of the automorphism group of the free group on x_{1}, y_{1}, ⋯, x_{g}, y_{g} that consists of the automorphisms fixing the fundamental relator R = [x_{1}, y_{1}][x_{2}, y_{2}] ⋯ [x_{g}, y_{g}]. The binary operation on ⨿_{g≥1}Γ_{g,1} induced by the F -product can be identified with the operation taking the free product of the automorphisms. The (r, s)-braiding on the free group on x_{1}, y_{1}, ⋯, x_{g}, y_{g} can be expressed by: