A PRESENTATION OF MAPPING CLASS GROUPS IN TERMS

OF ARTIN GROUPS AND GEOMETRIC MONODROMY OF SINGULARITIES

MAKOTO MATSUMOTO

Abstract. From Wajnryb’s presentation, we extract a simple presentation of the mapping class group of the genus g surface as a quotient of an Artin group by simple relations among the centers of sub-Artin groups.

Topological meanings are given by using deformation of simple singularities.

Contents

1. A presentation of mapping class groups by Artin groups 1 1.1. Artin groups 1 1.2. Mapping class groups 2 1.3. Chain and lantern relations by Artin groups 5 1.4. Other relations 7 2. Geometric monodromy of simple singularities 8 2.1. Topology 8 2.2. Geometric monodromy of simple singularities 9 2.3. Calculation of the monodromy of the center 10 2.4. The relations between the center and the boundary Dehn twists 11 References 13

1. A presentation of mapping class groups by Artin groups

1.1. Artin groups. Let n be a positive integer and let I be {1, 2, …, n}. By a Coxeter matrix, we mean a symmetric n × n matrix M = (m_{ij}) with m_{ij} being an integer ≥ 2 or ∞. Its Artin group is defined by generators a_{1}, ⋯, a_{n} and relations a_{i}a_{j}a_{i} ⋯ = a_{j}a_{i}a_{j} ⋯, where both sides are words of length m_{ij}, for each m_{ij} < ∞. If we add the relation a_{i}^{2} = 1 for each i, then we get the Coxeter group of M. In the following, we consider only the case where all m_{ij} are finite.

The Coxeter matrix can be conveniently described by a graph Γ, where the vertex set is I and two vertices i, j are joined by (m_{ij} − 2) edges (hence no loops but multiple edges are allowed; from now on only such graphs are considered). This coincides with the classical notation of Dynkin diagrams.

Conversely, any graph Γ yields a Coxeter matrix. Its Artin group is denoted by A(Γ). For example, if we denote by P_{n} a straight path consisting of n vertices with n − 1 edges, then A(P_{n}) is isomorphic to the braid group of n + 1 strings. Let Γ be any connected simple (i.e. no multiple edges) graph. It is easy to see that the abelianization of A(Γ) induces a natural surjection

Date: 1997 July 28th.

deg : A(Γ) → ℤ,

which we call the degree of an element of A(Γ).

For an induced (i.e., every edge in Γ with its ends in H is also an edge of H, sometimes also called full) subgraph H of Γ, there exists an obvious homomorphism A(H) → A(Γ). We call A(H) the sub-Artin group associated with subgraph H, although there is no assurance for A(H) ⊂ A(Γ).

For general H, the structure of A(H) is not well understood. However, if H is a Dynkin diagram of classical root systems, namely, if H is one of A_{n}, B_{n}, D_{n} E_{6}, E_{7}, E_{8}, F_{4}, G_{2}, H_{3}, H_{4}, I_{2}(p), then the group structure is well analyzed by Brieskorn-Saito [7].

For the proof of the next theorem, see Sections 4–7 in [7], in particular Satz 7.1, Satz 7.2 and Zusatz below Satz 7.2. We say a word w of finite (possibly empty) product of a_{i} (i ⊂ I) is positive, if the word contains no a_{j}^{−1}.

Theorem 1.1 (Brieskorn-Saito). Let Γ be a Dynkin diagram. Consider the following properties of an element w ∈ A(Γ).

• w has a presentation as a positive word of a_{j}.

• For any a_{i}, a_{i}^{−1}w has a positive word presentation of a_{j}’s. Then, there exists a unique element Δ(Γ) satisfying these properties, which is minimal in the sense that if w satisfies the above properties, then Δ(Γ)^{−1}w has a positive word presentation by a_{j}’s. Δ(Γ) has the following properties.

(i) Δ(Γ) is mapped to the longest length element w_{0} in the Coxeter group and its degree is the same as the length of w_{0} (i.e. the number of positive roots).

(ii) The center of A(Γ) is free cyclic with a generator c(Γ), defined by c(Γ) = Δ(Γ)^{2} = Π^{h} if w_{0} ≠ −1 and c(Γ) = Δ(Γ) = Π^{h/2} if w_{0} = −1, where h is the Coxeter number and Π is a product of all a_{j}’s with an arbitrary order.

They also obtained a Garside-type[8] normal-form theorem. See Section 6, in particular Satz 6.6 in [7].

1.2. Mapping class groups. Let Σ_{g}^{n,<b>} denote a compact oriented genus g surface with n ordered points specified and with b boundary components.

Its mapping class group M_{g}^{n,<b>} is defined to be the group of isotopy classes of orientation preserving self-diffeomorphisms of Σ_{g}^{n,<b>} which fix the n points pointwise, and are identity on the boundary.

We denote Σ_{g}^{<b>} := Σ_{g}^{0,<b>}, Σ_{g}^{n} := Σ_{g}^{n,<0>}, Σ_{g} := Σ_{g}^{0,<0>}, and similarly for corresponding mapping class groups. These groups were proved to be finitely presented[16]. The explicit presentations of M_{g}^{<1>}, M_{g} were given by Hatcher-Thurston[9], Harer[10], and finally by Wajnryb[18]. The result is the following theorem [18] (see the notation in [2]). Let C_{0}, C_{1}, …, C_{2g} be the simple closed curves on Σ_{g}^{<1>} as in Figure 1. Let c_{0}, c_{1}, …, c_{2g} denote the corresponding Dehn twists in M_{g}^{<1>}.

Theorem 1.2 (Wajnryb). The mapping class group M_{g}^{<1>} admits a presentation with generators a_{0}, a_{1}, …, a_{2g}, which are mapped to c_{0}, c_{1}, …, c_{2g}, and relations:

(A): a_{i}a_{j} = a_{j}a_{i} if C_{i} ∩ C_{j} = ∅, and a_{i}a_{j}a_{i} = a_{j}a_{i}a_{j} if C_{i} ∩ C_{j} consists of one point.

Figure 1. Dehn-Lickorish-Humphries Generators of Mapping Class Groups

Figure 2. The graph T_{g}

(B): (For g ≥ 2.) (a_{1}a_{2}a_{3})^{4} = a_{0}(a_{4}a_{3}a_{2}a_{1}a_{1}a_{2}a_{3}a_{4})^{−1}a_{0}(a_{4}a_{3}a_{2}a_{1}a_{1}a_{2}a_{3}a_{4}). (C): (For g ≥ 3.)

a_{0}b_{1}b_{2} = a_{1}a_{3}a_{5}b_{3},

where

b_{1} := (a_{4}a_{3}a_{5}a_{4})a_{0}(a_{4}a_{3}a_{5}a_{4})^{−1},
b_{2} := (a_{2}a_{1}a_{3}a_{2})b_{1}(a_{2}a_{1}a_{3}a_{2})^{−1},

b_{3} := (a_{4}^{−1}a_{3}^{−1}a_{2}^{−1}a_{1}^{−1}ua_{2}a_{3}a_{4}a_{5}a_{6})^{−1}a_{0}(a_{4}^{−1}a_{3}^{−1}a_{2}^{−1}a_{1}^{−1}ua_{2}a_{3}a_{4}a_{5}a_{6}), and

u := (a_{5}a_{6})b_{1}(a_{5}a_{6})^{−1}.

Here, a_{i} is mapped to c_{i} in the mapping class group. Moreover, we obtain the presentation of M_{g} by adding a relation (D) (omitted here, see [18]).

For the topological meanings of these relations, see Birman’s survey [2]. Because of the pictorial descriptions, the relations (A), (B), (C), (D) are called braid, chain, lantern, hyperelliptic relations, respectively. (Note that (D) in her survey has a small mistake in (7) on P.20., compare with Wajnryb’s correct one.)

Let T_{g} be the graph shown in Figure 2. It is clear from the relation (A) that we have a surjective homomorphism A(T_{g}) → M_{g}^{<1>}. It is natural to consider how the other relations can be interpreted in terms of Artin groups of finite types. It turns out that the relations (B), (C), (D) are equivalent to very simple relations among c(Γ) for several Γ. Let H be an induced subgraph of T_{g} of Dynkin type. Then, we denote by the same symbol c(H) the image of c(H) ∈ A(H) → A(T_{g}) in A(T_{g}). We consider only the following subgraphs: H = one of A_{4}, A_{5}, E_{6} and E_{7} as shown in Figure 3, or H = A′_{2g}, D_{2g−1} as in Figure 4.

4 MAKOTO MATSUMOTO

Figure 3. Sub-Artin groups whose centers give (b) and (c) Figure 4. Sub-Artin groups whose centers give (d) and (e)

Theorem 1.3. Under the relation (A), the relation (B) in Theorem 1.2 is equivalent to

(b) c(A_{5}) = c(A_{4})^{2}.

Under the relations (A) and (b), the relation (C) is equivalent to

(c) c(E_{7}) = c(E_{6}).

Under the relations (A), (b) and (c), the relation (D) is equivalent to

(d) a_{1}^{2g−2} = c(D_{2g−1}).

A presentation of M_{g}^{1} is obtained by the relations (A), (b), (c), and

(e) c(A′_{2g})^{2} = 1.

For the case g = 3, we may replace (d) with

(f) c(A_{6})Δ(E_{6}) = Δ(E_{6})c(A_{6}),

and (e) with

(g) c(E_{7}) = 1.

Thus, we have presentations A(T_{g})/[(b), (c)] ≅ M_{g}^{<1>}, A(T_{g})/[(b), (c), (e)] ≅ M_{g}^{1}, and A(T_{g})/[(b), (c), (d)] ≅ M_{g}.

Remark 1.1. Explicitly, we have the following.

c(A_{4}) := (a_{0}a_{2}a_{3}a_{4})^{5}
c(A_{5}) := (a_{0}a_{1}a_{2}a_{3}a_{4})^{6}
c(E_{6}) := (a_{0}a_{2}a_{3}a_{4}a_{5}a_{6})^{12}
c(E_{7}) := (a_{0}a_{1}a_{2}a_{3}a_{4}a_{5}a_{6})^{9}
c(A′_{2g}) := (a_{1}a_{2}a_{3}a_{4} ⋯ a_{2g})^{2g+1}
c(D_{2g−1}) := (a_{0}a_{3}a_{4}a_{5} ⋯ a_{2g})^{4g−4}
Δ(E_{6}) := (a_{0}a_{2}a_{3}a_{4}a_{5}a_{6})^{4}(a_{0}a_{2}a_{3}a_{4}a_{5})(a_{0}a_{2}a_{3}a_{4})(a_{2}a_{3}a_{4})

Here, c(Γ) does not depend on the order of the elements in the parenthesis.

Remark 1.2. For genus three, the new presentation by (b), (c), (f), (g) is useful to realize the Hecke algebra representation of the mapping class group (§10 in [14], where Jones succeeded in the genus two case). K. Nishiyama and the author are preparing an article for this.

Remark 1.3. Looijenga and the author are going to obtain a purely algebraic geometric proof (independent of Hatcher-Thurston) of the presentation for M_{3}^{1}, using the strata of the moduli space given in [15]. It seems to be an interesting future work to put these results together to give an independent proof of Theorem 1.3.

The topological meaning of these relations are as follows.

Theorem 1.4. The image of c(A_{4})^{2}, c(A_{5}), c(E_{6}), c(E_{7}) in M_{3}^{<2>} is, respectively, f_{2}, f_{0}f_{1}, f_{4}, f_{0}^{2}f_{3}, where f_{i} is the Dehn twists along F_{i} shown in Figure 5. The image of Δ(E_{6}) is the half twist along F_{4}, that is, rotate the genus three surface bounded by F_{4} anticlockwise 180^{∘} so that the left and right holes exchange positions.

Theorem 1.5. The image of c(A′_{2g}) is the hyperelliptic involution, and that of c(D_{2g−1}) is f_{5}^{2g−3}f_{6}, shown in Figure 6.

These theorems may be shown by direct calculation of Dehn twists, but here we shall use geometry of deformation of singularities, see §2.

1.3. Chain and lantern relations by Artin groups. The equivalence of (B) and (b), (C) and (c) in Theorem 1.3 is shown by a computer program implementing the Brieskorn-Saito Algorithm [7] of Garside type[8] to obtain their normal form. Here the transformation is briefly explained so that the reader may reproduce the calculation by using a computer.

Computational result for (B). The relation (B) can be written as

(a_{1}a_{2}a_{3})^{4}(a_{4}a_{3}a_{2}a_{1}a_{1}a_{2}a_{3}a_{4})^{−1}a_{0}^{−1}(a_{4}a_{3}a_{2}a_{1}a_{1}a_{2}a_{3}a_{4})a_{0}^{−1} = 1.

The computer program showed that the normal form of the conjugate of the left hand side

(a_{1}a_{2}a_{3}a_{4})[(a_{1}a_{2}a_{3})^{4}(a_{4}a_{3}a_{2}a_{1}a_{1}a_{2}a_{3}a_{4})^{−1}a_{0}^{−1}(a_{4}a_{3}a_{2}a_{1}a_{1}a_{2}a_{3}a_{4})a_{0}^{−1}](a_{1}a_{2}a_{3}a_{4})^{−1}

is

Δ_{12340}^{−2}Δ_{2340}^{4},

(here Δ_{ijk···} denotes Δ(H) with H the induced subgraph by the vertices i, j, k, …,) i.e., this is equivalent to

c(A_{5}) = c(A_{4})^{2}.

6 MAKOTO MATSUMOTO

Figure 5. The image of centers of sub-Artin groups Figure 6. The image of centers of sub-Artin groups Computational result for (C). This is the hard part. First, note that (B) is equivalent to Δ_{12340}^{2} = Δ_{2340}^{4}, which can be rewritten as

a_{1}a_{2}a_{3}a_{4}a_{0}a_{0}a_{4}a_{3}a_{2}a_{1} = Δ_{2340}^{2},

since Δ_{12340}^{2} = a_{1}a_{2}a_{3}a_{4}a_{0}a_{0}a_{4}a_{3}a_{2}a_{1}Δ_{2340}^{2} (easy to check by drawing braids). After eliminating b_{1}, b_{2}, b_{3}, u from (C), put

L := [the left hand side of (C)][the right hand side of (C)]^{−1}.

Let W be the positive word which is a product of the following 132 a_{j}’s, whose suffix is in the order

53042 33425 30645 30642
53042 53106 42534 00453
64253 10425 30642 53064
53042 33425 31064 25306
42531 06425 31064 25306
45306 42313 42530 45306
42531 42304 56.

This word was found by trial and error. In the following, we shall denote simply j instead of a_{j}, for such long words. Then,

W LW^{−1}

has its normal form

Δ_{0123456}^{−1}K,

where K turns out to be T_{1}T_{2}T_{3} with

T_{1} = (5304232342530645306425304253),
T_{2} = (06456123400432156453),
T_{3} = (04253064253064).

Then we replace 1234004321 in T_{2} by Δ_{2340}^{2} by using (B), to obtain T′_{2}. Then, the normal form of T_{1}T′_{2}T_{3} turns out to be Δ_{023456}^{2}. Thus, under the relation (b), it is proved that (C) is equivalent to Δ_{0123456}^{−1}Δ_{023456}^{2} = 1, i.e., to (c).

1.4. Other relations. Since it is proven that A(T_{g})/[(b), (c)] ≅ M_{g}^{<1>}, we may use the topological relations among Dehn twists for other relations under (b) and (c). The next is well known.

Lemma 1.1. Let c be the Dehn twist along a curve C, and let σ ∈ M_{g}^{n,<b>}. Then, The conjugate σcσ^{−1} is the Dehn twist along σ(C).

Lemma 1.2. The kernel of M_{g}^{<1>} → M_{g}^{1} is generated by the Dehn twist f_{∞}, and the kernel of M_{g}^{<1>} → M_{g} is normally generated by d_{g}^{−1}d′_{g} shown in Figure 7. Thus, the relation (D) is equivalent to d_{g} = d′_{g}.

Proof. If σ is in the kernel of the first map, then σ can be isotopically deformed to be identity on the left side of f_{∞}. Then, σ is a mapping class group of the annulus, which is known to be generated by the Dehn twist f_{∞}. The latter statement follows from that the kernel of M_{g}^{1} → M_{g} is π_{1}(Σ_{g}), and is generated by the move of the marked point, or the difference of a pair of Dehn twists which separates this point. See Wajnryb[18] for the detail.

Figure 7. Curves for the relation (D).

Proof of the rest of the Main Theorem. Note that (f_{5}, f_{6}) in Figure 7 is, as a pair of curves, the image of (d_{g}, d′_{g}) by a mapping class, so by Lemma 1.1 it suffices to prove that (d) is equivalent to f_{5} = f_{6}. If we plug the last picture of Figure 6 into Figure 7, then the left boundary is filled and c_{1} = f_{5}. Now (d) is equivalent to f_{5} = f_{6}, and by Theorem 1.5 it is equivalent to f_{5}^{2g−2} = f_{5}^{2g−3}f_{6} = c(D_{2g−1}). This proves the equivalence between (D) and (d).

Since c(A′_{2g})^{2} is the Dehn twist f_{∞}, the relation (e) gives M_{g}^{1} by Lemma 1.2. Suppose g = 3. Then c(E_{7}) = c(A′_{6})^{2} by Theorems 1.4 and 1.5. It is obvious

that Δ(E_{6}) and c(A′_{6}) commute in M_{3}, being horizontal and vertical involutions. It suffices to prove that d_{3} = d′_{3} is derived from this commutativity. For g = 3, d_{3} is Δ(E_{6})a_{1}Δ(E_{6})^{−1} and hence d′_{3} = c(A_{6})d_{3}c(A_{6})^{−1}, by Lemma 1.1. Thus the problem is reduced to the commutativity of c(A_{6}) with d_{3} = Δ(E_{6})a_{1}Δ(E_{6})^{−1}, which is a direct consequence of the assumption and the commutativity of c(A_{6}) and a_{1}.

Now, Theorem 1.3 is proved, assuming Theorems 1.4 and 1.5. In the next section, we prove these theorems.

2. Geometric monodromy of simple singularities

2.1. Topology. An Artin group with 2 ≤ m_{ij} ≤ 3 is called small. We shall show that a homomorphism from a small Artin group to M_{g}^{n,<b>} that maps each generator to a Dehn twist has a simple geometric configuration. This topological result is interesting by itself, and will be used to prove that the geometric monodromy action on the Milnor fiber coincides with those given by the Dehn twists along the vanishing cycles. This is classically known as the action on cohomology, and could be directly proved by using A’Campo’s real cut theory[1], but here we give an elementary independent proof, considering a possible future application for higher genus cases.

A key lemma is the following topological statement by Ishida[12].

Theorem 2.1. Let C, C′ be two closed curves in a surface Σ_{g}^{b,<n>}, homotopically shifted so that the number of intersections is minimized. Let c, c′ be the corresponding two Dehn twists. Then,

(i) c and c′ commute if and only if C ∩ C′ = ∅, (ii) c and c′ satisfy the braid relation cc′c = c′cc′ if and only if C ∩C′ = one point, (iii) c and c′ satisfy no relation, i.e., generate a free group, otherwise.

Then, the following proposition immediately follows.

Proposition 2.1. Let Γ be a simple graph. Suppose that there exists a group homomorphism

A(Γ) → M_{g}^{n,<b>},

which maps each generator a_{i} to a Dehn twist c_{i}. Take a simple closed curve C_{i} which represents c_{i} for each i, so that the number of the geometric intersections is minimal. (For example, take the geodesics.)

Then, C_{i} ∩ C_{j} = ∅, one point, respectively holds, for a_{i} and a_{j} are nonadjacent, adjacent, respectively.

There are several homeomorphic types for the tubular neighborhood N_{Γ} of the union of C_{j}. However, if Γ is acyclic, then N_{Γ} is unique up to homeomorphism. This can be shown as follows. Take one curve, say C_{1}. Then its tubular neighborhood is a ribbon C̃_{1}. Since it is orientable, its homeomorphic type is uniquely determined. Take another curve which crosses with C_{1} at one point, say C_{2}. A part of C_{2} appears on C̃_{1}. There is a unique way to fatten C_{2} such that its orientation is compatible with C_{1}. This can be iterated until all curves are fattened, since there is no cycle in Γ and every time when a curve is fattened, there is only one point intersecting with already fattened curves. Thus, we obtain

Proposition 2.2. Suppose Γ is acyclic in Proposition 2.1. Then, the tubular neighborhood N_{Γ} of the union of C_{i}’s is uniquely determined up to homeomorphism.

2.2. Geometric monodromy of simple singularities. Brieskorn[5] gave an explicit construction of the miniversal deformation of a simple singularity on a curve.

Theorem 2.2. Let Γ be a Dynkin diagram of type A_{n}, D_{n}, E_{n}, with rank n. Let T = Specℝ[z_{1}, …, z_{n}] be the associated n-dimensional affine space, W ⊂ GL(T) the Weyl group (i.e. the Coxeter group). Let P_{1}, …, P_{n} be the polynomial generator of the invariant ring ℝ[z_{1}, …, z_{n}]^{W}. Then, there exists a miniversal deformation

F → T^{ℂ}/W = Specℂ[P_{1}, …, P_{n}],

defined by a weighted homogeneous polynomial

F (x, y, P_{1}, …, P_{l}).

Brieskorn also showed that the fundamental group of the smooth-fiber locus in T^{ℂ}/W is the corresponding Artin group [6].

Theorem 2.3. Let Φ ⊂ T be the set of roots. For a root α ∈ Φ, let τ_{α} ∈ W be the reflection with respect to α, and H_{α} (α ∈ Φ) the reflection hyperplane. Fix a fundamental root system Π and let C be the corresponding Weyl chamber. Since C is contractible, we can take C as a base point of the fundamental group. Then,

A(Γ) ≅ π_{1}((T − ⋃_{α∈Φ} H_{α})^{ℂ}/W, C)

holds, if each a_{α}, α ∈ Π, is mapped to the path from C to τ_{α}(C), which goes 180^{∘} around the hyperplane H_{α}^{ℂ} adjacent to C. (I.e., if we take the quotient of (T − H_{α})^{ℂ} by the translation along H_{α}, then we get a projection to ℂ − {0}. Under this projection, C is projected to the positive real line and τ_{α}(C) is projected to the negative real line. Take a path from the positive line to the negative line given by e^{π\sqrt{−1}t}, t : 0 → 1. Lift this path, then divide by W to obtain a closed path. )

Let us take a Milnor fiber F_{x}, with x ∈ C. (Here we say by a Milnor fiber the intersection of the fiber on x with a ball centered at the origin in the total space.) Then, the fundamental group of the smooth-fiber locus with base point x, i.e., A(Γ), acts on F_{x} modulo isotopy. This yields a morphism called geometric monodromy ρ_{Γ} : A(Γ) → M_{g}^{<b>}, where (g, b) is the homeomorphic type of the Milnor fiber. It is well known that a_{α} is a Dehn twist. Thus, by Proposition 2.2, we can prove the following.

Theorem 2.4. The geometric monodromy ρ_{Γ} is obtained by mapping each generator of A(Γ) to the Dehn twist along the corresponding vanishing cycle.

If we take the abelianization

A(Γ) → M_{g}^{<b>} → M_{g} → AutH^{1}(Σ_{g}),

then we get the well-known Picard-Lefchez formula.

Proof. By Proposition 2.2, from the Dynkin diagram one can reproduce the tubular neighborhood of the union of the C_{i}. Assume Γ = A_{k}. If k is odd, then it is well known that the Milnor fiber F_{Γ} is homeomorphic to Σ_{g}^{<2>}, where g = (k − 1)/2, and if k is even, then homeomorphic to Σ_{g}^{<1>}, where g = k/2. It is easy to check by induction that the tubular neighborhood N_{Γ} of C_{1}, …, C_{k} given in Proposition 2.2 is homeomorphic to F_{Γ}. If k = 1, then obviously the neighborhood is Σ_{0}^{<2>}, and for k = 2, then we add one ribbon to this, yielding Σ_{1}^{<1>}, and so on.

Thus, the monodromy ρ_{Γ} is determined if we know how N_{Γ} is embedded in F_{Γ}. Since the genus is the same, we may consider only the boundary of N_{Γ}. If one boundary component of N_{Γ} is contractible in F_{Γ}, then the other boundary component of N_{Γ} cuts off the two boundary components of F_{Γ}. This means that any set of vanishing cycles does not separate the two boundary components of the Milnor fiber, which is a contradiction. (C.f. The calculation of the monodromy of the center, independently given in §2.4, manifests this.)

Thus, each boundary component of N_{Γ} cuts off one boundary component of F_{Γ}, i.e., the embedding is homotopically isomorphic. This proves Theorem 2.4 for the A_{k} case, since obviously the theorem is true if F_{Γ} = N_{Γ}.

Other cases follow by the same argument, by checking the following two points:

(1) Tubular neighborhood is homeomorphic to the Milnor fiber. (2) The boundary components of the Milnor fiber are separated in the complement of N_{Γ}.

2.3. Calculation of the monodromy of the center. Here we follow the notation in Theorems 2.2 and 2.3. The following is well known (c.f. [4]).

Lemma 2.1. Let f_{α}(z_{1}, …, z_{n}) be the linear polynomial corresponding to the root α, and put

D := ( ∏_{α∈Φ^{+}} f_{α})^{2},

where Φ^{+} is the set of positive roots. Then, D defines

(T − ⋃ H_{α})^{ℂ}/W → ℂ − {0},

and by taking π_{1} we get the degree map.

Proof. Obviously D is invariant by W. By the definition of τ_{α}, its image by D is a circle around the origin.

Lemma 2.2. Let w_{0} be the longest length element in W, i.e., w_{0} maps C to −C. Let Δ′ be a path from C to −C in T^{ℂ}, given by

t ∈ [0, 1] → (e^{π\sqrt{−1}t}z_{1}, …, e^{π\sqrt{−1}t}z_{n}) ∈ T^{ℂ}.

Then,

Δ′ ∈ π_{1}((T − ⋃ H_{α})^{ℂ}/W, C)

satisfies

Δ′a_{α} = a_{w_{0}(α)}Δ′.

Proof. Since T^{ℂ}, H_{α}^{ℂ} has ℂ^{∗}-action, we have a homotopy

e^{π\sqrt{−1}t}a_{α}(s) : [0, 1] × [0, 1] → (T − ⋃ H_{α})^{ℂ}.

Since −a_{α}(s) defines a path from −C to −τ_{α}(C) along the root w_{0}(α), after divided by W, we obtain the desired formula.

Proposition 2.3.

Δ′ = Δ(Γ) ∈ A(Γ).

Proof. By [7],

Δ(Γ)a_{α} = a_{w_{0}(α)}Δ(Γ)

holds. Thus, Δ′Δ(Γ)^{−1} lies in the center. Since (Δ′)^{2} is given by t : 0 ↦ 2 in Lemma 2.2, its degree is the same as the degree of D as a homogeneous polynomial, i.e., twice the number of positive roots. Thus, the degree of Δ′ is the number of positive roots, the same as Δ(Γ). Since the degree map restricts to an injection on the center, we have the result.

2.4. The relations between the center and the boundary Dehn twists. Brieskorn[5] gave the weight of the variables of F (x, y, P (z)). Let w_{1}, w_{2} be the weight of x, y, respectively. Each fiber is naturally compactified by filling one, two, or three points at infinity. We can choose a local parameter u_{j} of the form x^{n_{j}}y^{m_{j}}, n_{j}, m_{j} ∈ ℤ, at each point of infinity ∞_{j} (j = 1, 2, 3). This is a homogeneous polynomial with weight w_{∞j} = n_{j}w_{1} + m_{j}w_{2}.

By Proposition 2.3, a diffeomorphism of the fiber representing Δ(Γ) is given in the top arrow in the next diagram, where t is deformed from 0 to 1. More precisely, fix z = (z_{1}, …, z_{n}) to be a constant point in the (real) Weyl chamber, then move x, y smoothly (here there is choice of the move) according to the move of t, so that the boundary is isotopically fixed.

\begin{matrix}F (x, y, P (z)) & ⟶^{F (x,y,P (e^{π\sqrt{−1}t}z))} & F (x, y, P (−z)) \\ || & & ↑ isom. \\ F (X, Y, P (Z)) & ⟶^{F (e^{−w_{1}π\sqrt{−1}t}X,e^{−w_{2}π\sqrt{−1}t}Y,P (Z))} & F (e^{−w_{1}π\sqrt{−1}}X, e^{−w_{2}π\sqrt{−1}}Y, P (Z)). \end{matrix} (2.1)

The top deformation can be simulated by an isotopy

F (x, y, P (e^{π\sqrt{−1}t}z)) ≅ F (e^{−w_{1}π\sqrt{−1}t}X, e^{−w_{2}π\sqrt{−1}t}Y, P (Z))

given by X = e^{w_{1}π\sqrt{−1}t}x, Y = e^{w_{2}π\sqrt{−1}t}y, and Z = e^{π\sqrt{−1}t}z, along t : 0 → 1, and finally by mapping

F (e^{−w_{1}π\sqrt{−1}}X, e^{−w_{2}π\sqrt{−1}}Y, P (Z)) → F (x, y, P (−z))

by x = e^{−w_{1}π\sqrt{−1}}X, y = e^{−w_{2}π\sqrt{−1}}Y, and z = −Z.

In the lower deformation of (2.1), each fiber is isotopically invariant. Nontrivial monodromy appears only if we cut out neighborhood of the points at infinity, i.e., appears at the boundary. Each boundary is the image of |u_{j}| = ε for a positive real ε. A local parameter at the point at infinity ∞_{j} which is invariant in the bottom deformation is

U_{j}(t) := (e^{−w_{1}π\sqrt{−1}t}X)^{n_{j}}(e^{−w_{2}π\sqrt{−1}t}Y)^{m_{j}} = e^{−(n_{j}w_{1}+m_{j}w_{2})π\sqrt{−1}t}u_{j}.

To fix the boundary in the top deformation, we fix u_{j}, and thus the bottom parameter U_{j} is twisted by −(n_{j}w_{1} + m_{j}w_{2})π = −w_{∞j}π radian, and this gives the deformation at the bottom in (2.1). Then, we compose with this diffeomorphism an isomorphism given by x = e^{−w_{1}π\sqrt{−1}}X, y = e^{−w_{2}π\sqrt{−1}}Y, and z = −Z. Thus, we get the following

Proposition 2.4. The geometric monodromy of Δ(Γ) is obtained as follows. (i) Twist each boundary component j by −w_{∞j}π radian. (ii) Compose with the isomorphism x = e^{−w_{1}π\sqrt{−1}}X, y = e^{−w_{2}π\sqrt{−1}}Y.

The explicit monodromy is easily obtained from the following table. Here, we numbered the boundary components so that 1 is assigned to the boundary component on the smooth component ℙ^{1} in the degenerated fiber, if it exists. The weights are copied from the table in [5], and the other values follow by a simple calculation of the ramification indices.

Table I. Weight at infinities of simple singularities

Type  A_{n}  D_{n}
equation  x^{n+1} + y^{2}  x(x^{n−2} + y^{2})
w_{1}, w_{2}  (1, (n + 1)/2)  (2, n − 2)
n=even  n=odd  n=even  n=odd
# of components  1  2  3  2
−(w_{∞1}, w_{∞2}, w_{∞3})  (w_{1}/2)  (w_{1}, w_{1})  (w_{2}, w_{1}, w_{1})  (w_{2}, w_{1}/2)
(1/2)  (1, 1)  (n − 2, 2, 2)  (n − 2, 1)

Description of Δ

Boundary twists  π/2 rad.  π, π  (n − 2)π, 2π, 2π (n − 2)π, π
Image of (x, y)  −ix, (−1)^{(n+1)}y  −x, −y  identity  x, −y
c(Γ)  hyp. ell. inv. (1,1) twists  as above  (n-2,1) twists
Type  E_{6}  E_{7}  E_{8}
equation  x^{4} + y^{3} y(x^{3} + y^{2}) x^{5} + y^{3}
w_{1}, w_{2}  (3,4)  (4,6)  (6,10)
# of components  1  2  1
(w_{∞1}, w_{∞2}, w_{∞3}) (w_{1}/3) (w_{1}, w_{1}/2) (w_{1}/3)
(1) (4, 2) 2

Description of Δ

Boundary twists  π  4π, 2π  2π
Image of (x, y)  −x, y  identity identity
c(Γ)  1 twist as above as above

In particular, this table gives the proof of Theorems 1.4 and 1.5.

For example, we shall prove the latter theorem. Consider D_{n} with n odd. The corresponding singular curve is the union of the smooth line x = 0 and a cusped line x^{n−2} + y^{2} = 0. The picture of Dehn twists is as in Figure 8. Now we see the boundary component 1 came from the boundary along the smooth line, by observing that two vanishing cycles cut out one pants at the left. (More precisely, if we deform only the cusped line to one-punctured hyperelliptic curve, then the

Figure 8. Configuration of D_{n}, n odd

one-punctured projective line intersects with this curve normally at two points. This shows two vanishing cycles cut out one pants with the desired boundary 1.) The local parameter corresponding to the component 1 is y^{−1}, hence its weight is −w_{2} = −(n − 2). For the component 2, the local parameter is ramified with index two above x^{−1}. Thus it has weight −w_{1}/2 = −1. Since Δ(Γ) maps y to −y, its monodromy is the vertical involution with twists at boundaries by (n − 2)π, π radian, anticlockwise, so that the boundary is fixed. Then, c(Γ) corresponds to the products of the Dehn twists on the boundary with these weights, i.e., D_{1}^{n−2}D_{2}. By putting n = 2g − 1, we get Theorem 1.5.

Remark 2.1. It is interesting to observe that some of these center-boundary relations also played important roles in classical literatures. For example, the chain relation (c.f. [2]) is exactly same with the relation for c(A_{3}). Humphries’ relation[11] used to get the minimal generating set is essentially equivalent to the relation about Δ(E_{6}). See the end of §1.4, where Δ(E_{6}) maps a_{1} to d_{3}.

Acknowledgments. I would like to show my sincere gratitude to the following persons, who kindly helped this research: K. Saito for informing me of the Artin group and its relation to deformation of singularities, E. Brieskorn and P. Slodowy for fundamental facts on singularities, E. Looijenga for the relation between the moduli of genus three curves and E_{7} Artin groups, Takayuki Oda for the motivation and the basics on mapping class groups. This research is an extension of the first section of [17], which was motivated by a joint work with Ihara[13], to whom I am grateful.

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Max Planck Institut Fur Mathematik, Gottfried-Claren-Strasse 26 D-53225 BONN, GERMANY. (Until Sept. 1997). Department of Mathematics, Keio University, 3-14-1 Hiyoshi Kohoku-ku Yokohama 223 Japan.

   E-mail address: matumoto@math.keio.ac.jp