HOMOLOGICAL ACTION OF THE MODULAR GROUP ON SOME CUBIC MODULI SPACES

WILLIAM M. GOLDMAN AND WALTER D. NEUMANN

Abstract. We describe the action of the automorphism group of the complex cubic x^{2} + y^{2} + x^{2} − xyz − 2 on the homology of its fibers. This action includes the action of the mapping class group of a punctured torus on the subvarieties of its SL(2, ℂ) character variety given by fixing the trace of the peripheral element (socalled “relative character varieties”). This mapping class group is isomorphic to PGL(2, ℤ).

We also describe the corresponding mapping class group action for the four-holed sphere and its relative SL(2, ℂ) character varieties, which are fibers of deformations x^{2} + y^{2} + z^{2} − xyz − 2 − P x − Qy − Rz of the above cubic. The 2-congruence subgroup PGL(2, ℤ)_{(2)} still acts on these cubics and is the full automorphism group when P, Q, R are distinct.

Contents

Introduction 1 1. Vanishing cycles and homology 5 2. Action of Γ 7 3. Intersection form 9 4. Topology of fibers via the 27 lines on a cubic 10 5. The four-holed sphere 14 References 19

Introduction

Several important moduli spaces are described by cubic surfaces in affine space. The symmetries of the original moduli problem determines a group of transformations of the moduli space which generates a dynamical system. For two bounded surfaces M — the one-holed torus

Research supported under NSF grant DMS-0103889. Research supported under NSF grants DMS-0083097 and xxxxxx.

and the four-holed sphere — the relative SL(2, ℂ)-character varieties are described by cubic surfaces M in ℂ^{3} of the form

x^{2} + y^{2} + z^{2} − xyz = P x + Qy + Rz + S

upon which the mapping class group π_{0} Homeo(M, ∂M) acts by polynomial automorphisms. For these surfaces π_{0} Homeo(M, ∂M) is essentially the modular group PGL(2, ℤ). We compute the homology of M and its intersection form, and the resulting action of PGL(2, ℤ) on H_{∗}(M).

A relative SL(2, ℂ)-character variety of a compact bounded surface M classifies equivalence classes of representations

ρ : π_{1}(M) ⟶ SL(2, ℂ)

with certain boundary conditions. Namely let ∂_{i}M for i = 1, …, n denote the boundary components of M and choose δ_{i} ∈ π_{1}(M) corresponding to generators of π_{1}(∂_{i}M) respectively and and t = (t_{1}, …, t_{n}) ∈ ℂ^{n}. Then the relative SL(2, ℂ)-character variety with boundary data t is the moduli space of representations ρ where

tr (ρ(δ_{i})) = t_{i}

for i = 1, …, N.

Suppose first that M is a one-holed torus. Its fundamental group π is free of rank two, and admits a redundant geometric presentation

⟨X, Y, δ_{1} | δ_{1} = XY X^{−1}Y^{−1}⟩

where δ_{1} corresponds to a generator of π_{1}(∂M). Fricke [6] proved that the character variety is an affine space ℂ^{3} with coordinates

x = tr(ρ(X))
y = tr(ρ(Y))
z = tr(ρ(Z)), Z = XY .

The resulting π_{0} Homeo(M, ∂M)-action on this ℂ^{3} factors through an action of PGL(2, ℤ). Namely, by Dehn (unpublished) and Nielsen [14], π_{0} Homeo(M, ∂M) ≅ Out(π) ≅ GL(2, ℤ) via the action on the abelianization ℤ^{2} of π = π_{1}(M). The center {±I} of GL(2, ℤ) is represented by the automorphism X ↦ X^{−1}, Y ↦ Y^{−1}, Z ↦ Y Z^{−1}Y^{−1}, which preserves traces and hence acts trivially on ℂ^{3}, so the action can be considered as an action of GL(2, ℤ)/{±I} = PGL(2, ℤ).

The relative character varieties result by fixing the trace of ρ(δ_{1}), which is given by the polynomial

tr(ρ(XY X^{−1}Y^{−1})) = κ(tr(ρ(X)), tr(ρ(Y)), tr(ρ(XY))

where

κ(x, y, z) := x^{2} + y^{2} + z^{2} − xyz − 2.

Thus, the action of PGL(2, ℤ) on ℂ^{3} preserves this polynomial κ(x, y, z).

The full automorphism group Γ = Aut(κ) of this polynomial κ is generated by PGL(2, ℤ) and the “group of sign changes,” namely the Klein 4-group Σ ≅ C_{2} ×C_{2} that acts on ℂ^{3} by (x, y, z) ↦ (±x, ±y, ±z) with an even number of minus signs. In fact (see Horowitz [10]) Γ is the split extension:

Γ = PGL(2, ℤ) ⋉ Σ.

The dynamic behavior of the action of Γ on the real character variety ℝ^{3} ⊂ ℂ^{3} was described in [8], but little is known about the full action on ℂ^{3}. In this note we investigate the algebraic topology of this action of Γ (and hence of PGL(2, ℤ)) on the fibers of the polynomial κ.

The results can be summarized:

Theorem 1. The fibers V_{t} := κ^{−1}(t) of the polynomial κ: ℂ^{3} → ℂ have reduced homology only in dimension 2. There are two special fibers: V_{2} and V_{−2} with homology H_{2}(V_{2}) ≅ ℤ and H_{2}(V_{−2}) ≅ ℤ^{4}, respectively. The generic fiber V_{t} with t ≠ ±2 has homology

H_{2}(V_{t}) = H_{2}(V_{−2}) ⊕ H_{2}(V_{2}) ≅ ℤ^{5}.

The automorphism group Γ acts on these homology groups via a homomorphism to S_{4} × C_{2}, with the symmetric group S_{4} acting by (ρ_{4}) ⊕ 1 on ℤ^{4} ⊕ ℤ, where ρ_{4} is the standard permutation representation of S_{4} on ℤ^{4}, and the cyclic group C_{2} acting by multiplication by ±1 on ℤ^{5}.

The mapping class group Out(π), acting on ℂ^{3} as the subgroup PGL(2, ℤ) of Γ, acts on the above homology groups as the subgroup S_{3} × C_{2} of S_{4} × C_{2}, where the S_{3} is the subgroup of S_{4} that fixes one of the generators of ℤ^{4}.

The homomorphism of Γ to S_{4}×C_{2} is as follows: The homomorphism to C_{2} is the determinant map

Γ = PGL(2, ℤ) ⋉ Σ → PGL(2, ℤ) ⟶^{det} C_{2}

The homomorphism to S_{4} arises via reduction modulo 2:

Γ = PGL(2, ℤ) ⋉ Σ ⟶ PGL(2, ℤ/2) ⋉ Σ = S_{3} ⋉ Σ ≅ S_{4}.

Here PGL(2, ℤ/2) ⋉ Σ ≅ S_{4} because PGL(2, ℤ/2) ⋉ Σ is the group of affine automorphisms of an affine plane 𝔸 = 𝔸^{2}(ℤ/2) and every permutation of the 4-element set 𝔸 is affine. The subgroup Σ is the group of translations, which acts simply transitively on 𝔸. The group S_{3} that fixes a chosen origin in 𝔸 and permutes the other three points is the group of linear automorphisms PGL(2, ℤ/2) = GL(2, ℤ/2) (since ℤ/2 contains only one nonzero scalar,

PGL(2, ℤ/2) = GL(2, ℤ/2) = PSL(2, ℤ/2) = SL(2, Z/2) ).

In terms of the punctured torus M, 𝔸 can be identified with the set of spin structures on a trivial vector bundle; specifically its vector space of translations identifies with the cohomology group H^{1}(M, ℤ/2).

Note that the kernel of Γ = PGL(2, ℤ)⋉Σ ⟶ PGL(ℤ/2, 2)⋉Σ = S_{4} is the congruence subgroup

PGL(2, ℤ)_{(2)} := \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ PGL(2, ℤ) | b ≡ c ≡ 0 mod 2\}

of PGL(2, ℤ). We have a commutative diagram with exact rows and columns

1  1
↓  ↓
Σ  ⟶^{=} Σ
↓  ↓
1 ⟶ PGL(2, ℤ)_{(2)} ⟶  Γ  ⟶ S_{4} ⟶ 1
↓^{=}  ↓  ↓
1 ⟶ PGL(2, ℤ)_{(2)} ⟶ PGL(2, ℤ) ⟶ S_{3} ⟶ 1
↓  ↓
1  1

The right-hand column identifies with the short exact sequence

1 ⟶ ℤ/2 ⊕ ℤ/2 ⟶ Aff(2, ℤ/2) ⟶ PGL(2, ℤ/2) ⟶ 1

All four short exact sequences of the diagram are split exact. In particular,

(1)  Γ ≅ PGL(2, ℤ)_{(2)} ⋊ S_{4} .

We now describe the case of the four holed sphere N. The group Γ arises in a new context in this discussion, as the mapping class group π_{0} Homeo(N, ∂N). The relative SL(2, ℂ) character varieties for N are again certain cubic surfaces. We first discuss the automorphisms of these cubic surfaces.

Theorem 2. For any P, Q, R ∈ ℂ, the action of PGL(2, ℤ)_{(2)} on ℂ^{3} that preserves the polynomial κ deforms to an action that preserves the cubic polynomial

κ_{P,Q,R}(x, y, z) := x^{2} + y^{2} + z^{2} − xyz − P x − Qy − Rz − 2 .

Each nonsingular fiber V of κ_{P,Q,R} is diffeomorphic to a non-singular fiber of κ, so it still has homology H_{2}(V) ≅ ℤ^{5}, while a singular fiber has homology a quotient of this. The action of PGL(2, ℤ)_{(2)} on the homology of fibers of κ_{P,Q,R} is again by multiplication by ±1 via the determinant homomorphism PGL(2, ℤ)_{(2)} → C_{2} = {±1}.

PGL(2, ℤ)_{(2)} is the full automorphism group of κ_{P,Q,R} if P, Q, R are distinct, while otherwise the full automorphism group is PGL(2, ℤ)_{(2)} extended by an appropriate subgroup of S_{4} (those permutations and sign changes of x, y, z that preserve κ_{P,Q,R}; — if at most one of P, Q, R is zero then there are only permutations).

Each relative character variety for N is a fiber of some cubic polynomial κ_{P,Q,R}(x, y, z). The mapping class group that fixes all four boundary components satisfies

π_{0} Homeo(N, ∂_{1}N, …, ∂_{4}N) ≅ PGL(2, ℤ)_{(2)} ,

and its action on the relative character variety is by the action of the above theorem.

If the trace constraints at some boundary components of M are identical, then the relevant mapping class group is PGL(2, ℤ)_{(2)} ⋉ K where K ⊂ S_{4} is the group of permutations that preserve trace constraints. Its action on the relative character variety is by the quotient PGL(2, ℤ)_{(2)} ⋉ π(K) where π: S_{4} → S_{3}. In particular, if the trace constraints at all four boundary components of N are equal, then the relevant mapping class group is a semidirect product PGL(2, ℤ)_{(2)} ⋊S_{4}. In fact it is the same semidirect product as that of equation (1), so

π_{0} Homeo(N, ∂N) ≅ Γ .

But its action on ℂ^{3} in Theorem 2 is via its homomorphism to PGL(2, ℤ), with kernel Σ, while the action of Γ on ℂ^{3} of Theorem 1 was faithful.

We also determine the intersection form on homology of fibers:

Proposition 3. The intersection form on the homology H_{2}(V_{t}) of a general fiber of any one of the cubics κ_{P,Q,R} with respect to the basis used in the above theorems is

(\begin{matrix}−2 & 0 & 0 & 0 & 1 \\ 0 & −2 & 0 & 0 & 1 \\ 0 & 0 & −2 & 0 & 1 \\ 0 & 0 & 0 & −2 & 1 \\ 1 & 1 & 1 & 1 & −2 \end{matrix}) .

Acknowledgments. Goldman is grateful to J. Damon, L. Ein, H. King, A. Libgober and J. W. Wood for valuable conversations.

1. Vanishing cycles and homology

It is known (e.g., [13]) that the homology of a general fiber of a polynomial map κ comes from vanishing cycles of the singularities and vanishing cycles “at infinity.”

Broughton [2] shows that a polynomial has no vanishing cycles at infinity if it is “tame”, that is, |dκ| is bounded away from zero outside some compact set. Our polynomial κ is tame, in fact:

Lemma 1.1. |dκ| tends to infinity as |(x, y, z)| → ∞.

Proof.

dκ = (2x − yz)dx + (2y − xz)dy + (2z − xy)dz =: adx + bdy + cdz

We shall show that ∥(a, b, c)∥ bounded implies ∥(x, y, z)∥ bounded. Consider a sequence of points (x, y, z) going to infinity for which (a, b, c) stays bounded. If two coordinates, say x and y, stay bounded then c = 2z − xy tends to infinity, a contradiction. So at least two coordinates, say y and z, must approach infinity in the sequence. But for z ≠ ±2 the equations 2x = a + yz and 2y = b + xz imply

y = \frac{2b + az}{4 − z^{2}} ,

which approaches 0 as z → ∞, another contradiction. □

For a tame polynomial, the homology of the general fiber is given by the vanishing cycles of singularities, so we need to find the singularities of κ:

2x − yz = 2y − xz = 2z − xy = 0

implies the critical points of κ are:

(2) (2, −2, −2), (−2, 2, −2), (−2, −2, 2), (2, 2, 2), (0, 0, 0).

The first four singular points are on the fiber V_{2} while the singular point (0, 0, 0) is on V_{−2}. For any one of the singular points (2ϵ_{1}, 2ϵ_{2}, 2ϵ_{1}ϵ_{2}) with ϵ_{i} = ±1 we introduce local coordinates

x̂ = ϵ_{1}x − 2, ŷ = ϵ_{2}y − 2, ẑ = ϵ_{1}ϵ_{2}z − 2 .

In these local coordinates we have

κ(x, y, z) = x̂^{2} + ŷ^{2} + ẑ^{2} − 2x̂ŷ − 2x̂ẑ − 2ŷẑ − x̂ŷẑ.

The Hessian at (0, 0, 0) is 2I and at each of the other four singular points (using the above local coordinates) it is

2 (\begin{matrix}1 & −1 & −1 \\ −1 & 1 & −1 \\ −1 & −1 & 1 \end{matrix})

Since these Hessians are non-singular, we have a quadratic singularity at each point, so each singular point contributes a single vanishing cycle to the homology of other fibers of κ. We hence have:

H_{2}(V_{2}) ≅ ℤ generated by the vanishing cycle at (0, 0, 0),
H_{2}(V_{−2}) ≅ ℤ^{4} generated by the vanishing cycles at (±2, ±2, ±2),
H_{2}(V_{t}) ≅ ℤ^{5} otherwise, generated by all five vanishing cycles.

The vanishing cycle at a quadratic singularity is only defined up to sign; a self-isomorphism of the singularity whose derivative has negative determinant multiplies the cycle by −1. We will choose a basis of ℤ^{5} = ℤ^{4} ⊕ ℤ by choosing vanishing cycles at (2, −2, −2) and (0, 0, 0) and using the identity map between our chosen local coordinates x̂, ŷ, ẑ at the four singular points (±1, ±1, ±1) to choose the vanishing cycles at the other three singular points. We will order our basis according to the ordering of singular points in (2) above.

2. Action of Γ

The Klein 4-group Σ of sign-changes permutes the first four singular points (2) of κ by the regular permutation representation. It acts as the identity map in the local coordinates that we introduced above. It therefore simply permutes the four vanishing cycles corresponding to these points. Moreover, it fixes the singular point (0, 0, 0) and acts with positive determinant in local coordinates there, so it fixes the corresponding vanishing cycle. It hence acts on ℤ^{5} by the 4-group whose non-trivial elements are:

(\begin{matrix}0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) ,  (\begin{matrix}0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) ,  (\begin{matrix}0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) .

The group PGL(2, ℤ) is generated by PSL(2, ℤ) and the involution ({−1 0 \atop 0 1}). We first discuss the group PSL(2, ℤ). It is the free product of cyclic groups of orders 2 and 3 generated by

(\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) and (\begin{matrix}1 & −1 \\ 1 & 0 \end{matrix})

respectively. These are represented by automorphisms

α: X ↦ Y, Y ↦ X^{−1}
β: X ↦ XY, Y ↦ X^{−1} .

Recalling that an automorphism γ of π acts on characters by (γχ)(g) = χ ∘ γ^{−1}(g), the action of the above two automorphisms on characters are:

α_{∗}: (x, y, z) ↦ (y, x, xy − z)
β_{∗}: (x, y, z) ↦ (y, z, x)

respectively.

The map α_{∗} exchanges the first two singular points and fixes the other three. Its derivative acts with determinant 1 in the local coordinates at the singular points. It thus acts on the homology ℤ^{5} of a general fiber by:

(\begin{matrix}0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix})

The map β_{∗} of ℂ^{3} permutes the first three singular points of (2) above and fixes the other two. It also acts with determinant 1 in the local coordinates at the singular points. It thus acts on the homology ℤ^{5} of a general fiber by:

(\begin{matrix}0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix})

The Klein 4-group Σ and the elements α and β generate the permutation representation of S_{4} on the first four coordinates of ℤ^{5}, as promised in the theorem of the Introduction.

Finally we discuss the involution

(\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix}) ∈ PGL(2, ℤ).

The automorphism of π defined by

γ: X ↦ X^{−1}, Y ↦ Y

represents this involution, and induces the map

γ_{∗}: (x, y, z) ↦ (x, y, xy − z)

on the character variety ℂ^{3}. This fixes all the singular points of κ and its derivative acts with determinant −1 in the local coordinates at each point. It thus reverses the signs of all the vanishing cycles, so it acts on ℤ^{5} by the matrix

(\begin{matrix}−1 & 0 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 \\ 0 & 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 0 & −1 \end{matrix}) .

3. Intersection form

Consider a general fiber V_{t} (that is, t ≠ ±2). It is well known that the vanishing cycle of a quadratic singularity has self-intersection number −2. The vanishing cycles arising from the four singularities on V_{2} are clearly disjoint. Thus the intersection form on H_{2}(V_{t}) with respect to our basis is

(\begin{matrix}−2 & 0 & 0 & 0 & a_{1} \\ 0 & −2 & 0 & 0 & a_{2} \\ 0 & 0 & −2 & 0 & a_{3} \\ 0 & 0 & 0 & −2 & a_{4} \\ a_{1} & a_{2} & a_{3} & a_{4} & −2 \end{matrix}) ,

where a_{1}, …, a_{4} are still to be determined. Since an index 2 subgroup of Γ permutes the first four basis elements, a_{1} = a_{2} = a_{3} = a_{4}. Call this common value a. We must show a = ±1 (the sign depends on how we have oriented the vanishing cycles over t = 2 and t = −2 with respect to each other, and is thus indeterminate).

The intersection matrix has determinant 32(1 − a^{2}). Let V̂_{t} be the intersection of V_{t} with a very large ball around the origin and M^{3} = ∂V̂_{t} the “link at infinity” of V_{t}. Then V_{t} retracts to V̂_{t}, so they have the same homology and intersection form. The homology exact sequence

H_{2}(V̂_{t}) → H_{2}(V̂_{t}, M^{3}) → H_{1}(M^{3}) → 0

and Poincaré-Lefschetz duality

H_{2}(V̂_{t}, M^{3}) ≅ H^{2}(V̂_{t})) ≅ H_{2}(V̂_{t})^{∗}

yield a standard long exact sequence

H_{2}(V̂_{t}) → H_{2}(V̂_{t})^{∗} → H_{1}(M^{3}) → 0

where the first arrow is intersection form. Thus H_{1}(M^{3}) is the cokernel of the intersection matrix, so a = ±1 if and only if H_{1}(M^{3}) is infinite (and otherwise |H_{1}(M^{3})| would equal 32|1 − a^{2}|)). It thus remains to prove that H_{1}(M^{3}) is indeed infinite.

Consider M^{3} as a regular neighborhood boundary of the divisor at infinity of the closure \overline{V}_{t} of V_{t} in ℙ^{3}(ℂ) = ℂ^{3} ∪ ℂℙ^{2}. Since the highest order term of κ is xyz, \overline{V}_{t} intersects the projective plane at infinity in the union of the three lines x = 0, y = 0, and z = 0 (these are three of the 27 lines on a cubic surface; we will meet the other 24 lines on V_{t} in the next section). It is easy to check in local coordinates that \overline{V}_{t} is non-singular at infinity (and hence globally non-singular except when t = ±2). The three lines at infinity of V_{t} intersect in a cyclic configuration, so their neighborhood boundary certainly has infinite first homology, as desired.

In fact (see [12]), the neighborhood boundary M^{3} of any cyclic configuration of rational curves in a non-singular complex surface is a T^{2} bundle over S^{1} whose monodromy can be computed from the selfintersection numbers e_{i} of the curves as

A = ∏_{i} (\begin{matrix}0 & −1 \\ 1 & −e_{i} \end{matrix}) .

In our case it is not hard to calculate that each of the three curves has self-intersection −1 (this is true for all 27 lines on a cubic surface) so the monodromy is

(\begin{matrix}0 & −1 \\ 1 & 1 \end{matrix})^{3} = (\begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix})

and hence

H_{1}(M^{3}) = ℤ ⊕ (ℤ/2)^{2}.

One can check that this agrees with the cokernel of the intersection matrix of the proposition.

4. Topology of fibers via the 27 lines on a cubic

An alternative way of computing the homology of the fibers of κ and the action of Γ involves the projection to a coordinate line, and provides further insight into the topology. We sketch it briefly, hopefully with sufficient explanation that the reader can fill in the details. It is, in fact, the way we first obtained the results.

Suppose t ≠ ±2 and consider the projection π: V_{t} → ℂ given by the z-coordinate. The general fiber π^{−1}(s) of this projection is given by x^{2} + y^{2} − xys = t + 2 − s^{2}, which we can write as

(x − λy)(x − λ′y) = t + 2 − s^{2}

with λ, λ′ = \frac{1}{2}(s ± \sqrt{s^{2} − 4}). Thus π^{−1}(s) ≅ ℂ^{∗} by the map (x, y) ↦ x − λy unless λ = λ′ or t + 2 − s^{2} = 0. The general fiber of π is thus ℂ^{∗}, while the special fibers occur for s = ±2 and s = ±\sqrt{t + 2}. These special fibers are:

π^{−1}(2) = L_{1} ∪ L_{2}; L_{1} = \{x = y + \sqrt{t − 2}\},
L_{2} = \{x = y − \sqrt{t − 2}\}
π^{−1}(\sqrt{t + 2}) = L_{3} ∪ L_{4}; L_{3} = \{\frac{x}{y} = \frac{\sqrt{t+2}+\sqrt{t−2}}{2}\},
L_{4} = \{\frac{x}{y} = \frac{\sqrt{t+2}−\sqrt{t−2}}{2}\}
π^{−1}(−\sqrt{t + 2}) = L_{5} ∪ L_{6}; L_{5} = \{\frac{x}{y} = −\frac{\sqrt{t+2}+\sqrt{t−2}}{2}\},
L_{6} = \{\frac{x}{y} = −\frac{\sqrt{t+2}−\sqrt{t−2}}{2}\}
π^{−1}(−2) = L_{7} ∪ L_{8}; L_{7} = \{x = −y + \sqrt{t − 2}\},
L_{8} = \{x = −y − \sqrt{t − 2}\}

These 8 lines L_{1}, …, L_{8} and the analogous lines coming from the projections to the x and y coordinates give 24 lines, which with the 3 lines at infinity give the full 27 lines on the projective cubic \overline{V}_{t}.

Choose small disks in ℂ

D_{1}, D_{2}, D_{3}, D_{4} around z = 2, \sqrt{t + 2}, −\sqrt{t + 2}, −2,

and paths:

γ_{1} from 2 to \sqrt{t + 2},
γ_{2} from \sqrt{t + 2} to − \sqrt{t + 2},
γ_{3} from −\sqrt{t + 2} to − 2,

so that these paths do not intersect each other except at endpoints.

Then X := π^{−1}(D_{1} ∪ γ_{1} ∪ D_{2} ∪ γ_{2} ∪ D_{3} ∪ γ_{3} ∪ D_{4}) is a deformation retract of V_{t}. Moreover

π^{−1}(D_{1}) ≃ S^{2} ≃ π^{−1}(D_{4}) (see below)
π^{−1}(D_{2}) ≃ ∗ ≃ π^{−1}(D_{3}) (since π^{−1}(±\sqrt{t + 2}) ≃ ∗),

so a simple Mayer-Vietoris calculation of homology of X gives

H_{2}(V_{t}) ≅ ℤ ⊕ ℤ ⊕ ℤ ⊕ ℤ ⊕ ℤ
generators: α_{1}  α_{2}  α_{3}  α_{4}  α_{5}

where:

α_{1} is a generator of H_{2}(π^{−1}(D_{1}),
α_{5} is a generator of H_{2}(π^{−1}(D_{4}),
α_{2}, α_{3}, α_{4} are supported over γ_{1}, γ_{2}, γ_{3}.

We shall see that the intersection form for this basis is

(\begin{matrix}−4 & 2 & 0 & 0 & 0 \\ 2 & −2 & −1 & 0 & 0 \\ 0 & −1 & −2 & 1 & 0 \\ 0 & 0 & 1 & −2 & 2 \\ 0 & 0 & 0 & 2 & −4 \end{matrix})

To describe the α_{i} more explicitly we draw the special fibers and π^{−1}(D_{i}) in the projective completion \overline{V}_{t} of V_{t}:

Each line in the diagram has self-intersection number −1. We first describe the classes α_{2}, α_{3}, α_{4}.

For α_{2}, choose an arc γ in the line L_{0} at infinity from the point

L_{0} ∩ L_{2} = [1 : 1]

to the point

L_{0} ∩ L_{3} = [\frac{\sqrt{t + 2} + \sqrt{t − 2}}{2} : 1].

We can form the connected sum of L_{2} and −L_{3} by connecting them by a tube along this arc (the tube is s^{−1}(γ) where s: ∂NL_{0} → L_{0} is the projection of the boundary of a tubular neighborhood of L_{0}). This connected sum L_{2}# − L_{3} lies in V_{t} and represents the class α_{2}.

We construct α_{3} and α_{4} similarly as L_{4}# − L_{5} and L_{5}# − L_{7}. We shall see later that the choices of the tubes at infinity to form these connected sums does not affect the homology classes.

The classes α_{1} and α_{5} are constructed similarly as L_{1}# − L_{2} and L_{7}# − L_{8}, but it is not so clear how the tubes are chosen. To clarify this blow up the point L_{0} ∩ L_{1};

then π^{−1}(D_{1}) can be seen as the union of a portion N_{E} in a neighborhood of the exceptional curve E and portions N_{1} and N_{2} in neighborhoods of L_{1}, L_{2}. N_{E} is a (D^{2} − {0})-bundle over E − (E ∩ L_{0}), hence homotopy equivalent to a circle. Adding N_{1} and N_{2} to this just adds 2-handles along parallel copies of the circle, so π^{−1}(D_{1}) ≃ S^{2}, as claimed earlier. The S^{2} can be seen as L_{1}# − L_{2} with the connected sum formed along a tube over an arc in E. Its homology class α_{1} thus has self-intersection number

α_{1} · α_{1} = L_{1} · L_{1} + L_{2} · L_{2} = −2 + (−2) = −4

(since blowing up on L_{1} and L_{2} has decreased their self-intersection numbers by 1). The next paragraph gives an alternative argument.

We can see mutual intersection numbers of all our elements as follows. In the projective cubic \overline{V}_{t} an element such as α_{2} = L_{2}# − L_{3} is homologous to the sum L_{2}+(−L_{3}), since the tube used to perform connected sum bounds a solid tube. Since the inclusion V_{t} → \overline{V}_{t} respects intersection number, we can simply compute intersection numbers in \overline{V}_{t}. Thus, for instance, α_{2} clearly has self-intersection number

L_{2} · L_{2} + L_{3} · L_{3} = −2.

The claimed intersection matrix for the basis α_{1}, …, α_{5} is now easy to verify.

To see that the choices of the “tubes at infinity” do not affect the homology classes α_{2}, α_{3}, α_{4}, note that if we move the path in E_{0} defining such a tube past an intersection point E_{0} ∩ E_{i} then we change the homology class by the class of a torus s^{−1}(δ), where δ is a small loop in E_{0} around this point. Such a torus is supported entirely in π^{−1}(D_{i}) and has self-intersection number 0, so its homology class is zero.

One can verify that the ordered basis of section 1, corresponding to the vanishing cycles at

(2, −2, −2), (−2, 2, −2), (−2, −2, 2), (2, 2, 2), (0, 0, 0) ,

is

−(α_{4} + α_{5}), −α_{4},  α_{1} + α_{2},  α_{2},  −α_{3} .

In terms of our lines these are:

L_{8} − L_{5}, L_{7} − L_{5}, L_{1} − L_{3}, L_{2} − L_{3}, L_{5} − L_{4}.

Recall that we only determined the basis of section 1 up to an overall sign. If we replace each L_{i} by the other line in the same fiber of π, so L_{8} − L_{5} becomes L_{7} − L_{6} for example, each element of the basis is replaced by its negative.

The group Γ acts on the projective cubic \overline{V}_{t} by rational maps. It has a linear subgroup isomorphic to S_{4} (this subgroup is generated by the group of sign changes and the elements α_{∗}γ_{∗} and β_{∗} of section 2). This linear subgroup acts on \overline{V}_{t} by morphisms (biholomorphic maps) and therefore permutes the 27 lines. The whole of Γ is generated by this linear subgroup and the involution γ_{∗}. This latter involution acts on \overline{V}_{t} by the rational map which blows up the intersection point of the lines x = 0 and y = 0 at infinity and blows down the line z = 0 at infinity. It is not hard to derive the action on homology also from this point of view.

5. The four-holed sphere

Suppose N is a four-holed sphere. Its fundamental group π is freely generated by three peripheral elements δ_{1}, δ_{2}, δ_{3}. The generator δ_{4} of the fourth boundary component satisfies the relation

δ_{1}δ_{2}δ_{3}δ_{4} = 1

and we consider the redundant geometric presentation

⟨X, Y, Z, δ_{1}, δ_{2}, δ_{3}, δ_{4} | δ_{1}δ_{2}δ_{3}δ_{4} = 1, X = δ_{1}δ_{2}, Y = δ_{2}δ_{3}, Z = δ_{3}δ_{1}⟩.

Let t = (t_{1}, t_{2}, t_{3}, t_{4}) ∈ ℂ^{4}. The relative character variety results by fixing the trace

tr ρ(δ_{i}) = t_{i}

for i = 1, 2, 3, 4. In that case (see Goldman [7, 6], Benedetto-Goldman [5], which is based on Magnus [11]), the relative character variety is given by the cubic hypersurface in ℂ^{3} defined by:

x^{2} + y^{2} + z^{2} + xyz =(t_{1}t_{2} + t_{3}t_{4})x + (t_{1}t_{4} + t_{2}t_{3})y + (t_{1}t_{3} + t_{2}t_{4})z
+ (4 − t_{1}^{2} − t_{2}^{2} − t_{3}^{2} − t_{4}^{2} − t_{1}t_{2}t_{3}t_{4}).

Using the change of coordinates that reverses the signs of x, y and z puts this in the form:

κ_{P,Q,R}(x, y, z) = S

where

κ_{P,Q,R}(x, y, z) = x^{2} + y^{2} + z^{2} − xyz − P x − Qy − Rz − 2

as in the Introduction, and P = −(t_{1}t_{2} + t_{3}t_{4}), Q = −(t_{1}t_{4} + t_{2}t_{3}), R = −(t_{1}t_{3} + t_{2}t_{4}), and S = (2 − t_{1}^{2} − t_{2}^{2} − t_{3}^{2} − t_{4}^{2} − t_{1}t_{2}t_{3}t_{4}).

We first discuss the the group Aut(κ_{P,Q,R}) of polynomial automorphisms of the polynomial κ_{P,Q,R}.

Any affine automorphism ϕ = (ϕ_{1}, ϕ_{2}, ϕ_{3}) of ℂ^{3} that preserves κ_{P,Q,R} extends to ℙ^{3}(ℂ) and must therefore preserve the intersection of the closure \overline{V}_{t} of the fiber V_{t} = κ_{P,Q,R}^{−1}(t) of κ_{P,Q,R} with the plane at infinity. This intersection consists of the three coordinate lines. It follows that ϕ must permute the coordinates x, y, z up to multiplication by scalars. Inspecting κ_{P,Q,R} we see that the only non-trivial multiplication by scalars that can occur is multiplying two coordinates by −1 if the corresponding two of P, Q, R are zero. We thus see:

Lemma 5.1. The group L_{P,Q,R} = Aut(κ_{P,Q,R}) ∩ Aff(ℂ^{3}) is generated by permutation and sign-change automorphisms and is one of

{1}, C_{2}, C_{2} × C_{2}, S_{3}, S_{4},

according as no two of P, Q, R are equal; just two of them are equal but nonzero; just two of them are equal to zero; all three equal and nonzero; all three zero. □

Direct substitution shows that the following three involutions are in Aut(κ_{P,Q,R}):

τ_{1} : (x, y, z) ↦ (x, y, xy − z + R)
τ_{2} : (x, y, z) ↦ (yz − x + P, y, z)
τ_{3} : (x, y, z) ↦ (x, xz − y + Q, z) .

Theorem 5.2. Aut(κ_{P,Q,R}) is generated by the above three involutions and L_{P,Q,R}. The three involutions generate a normal subgroup which is the free product C_{2} ∗ C_{2} ∗ C_{2} = ⟨τ_{1}, τ_{2}, τ_{3} | τ_{1}^{2} = τ_{2}^{2} = τ_{3}^{2} = 1⟩ and

Aut(κ_{P,Q,R}) = (C_{2} ∗ C_{2} ∗ C_{2}) ⋊ L_{P,Q,R} .

(We will later identify this C_{2} ∗ C_{2} ∗ C_{2} with PGL(2, ℤ)_{(2)}.)

Proof. The proof follows Horowitz’s proof [10] for the case P = Q = R = 0. The group generated by the τ_{i} is normalized by L_{P,Q,R}. Let ϕ = (ϕ_{1}, ϕ_{2}, ϕ_{3}) ∈ Aut(κ_{P,Q,R}) have degree > 1. We claim there is a unique one of the three involutions τ_{i} for which τ_{i} ∘ ϕ has lower degree than ϕ. It then follows that there is a unique reduced word ω in the τ_{i} such that ω ∘ ϕ ∈ L_{P,Q,R}. This implies the theorem.

To prove the claim, assume for convenience that deg ϕ_{1} ≤ deg ϕ_{2} ≤ deg ϕ_{3} and deg ϕ_{3} > 1. The highest order terms in the equation

ϕ_{1}^{2}+ϕ_{2}^{2}+ϕ_{3}^{2}−ϕ_{1}ϕ_{2}ϕ_{3}−P ϕ_{1}−Qϕ_{2}−Rϕ_{3} = x^{2}+y^{2}+z^{2}−xyz−P x−Qy−Rz

imply that deg ϕ_{3} = deg ϕ_{1} + deg ϕ_{2} and that the highest order terms in ϕ_{1}ϕ_{2} − ϕ_{3} must cancel. It follows that τ_{1} ∘ ϕ has lower degree than ϕ. Clearly, neither of the other τ_{i} has this property. □

We now discuss the mapping class group π_{0} Homeo(N, ∂_{1}N, …, ∂_{4}N) which fixes the boundary components of the four-holed sphere N.

We can array the punctures of N in order around the equator of a 2-sphere and place our basepoint on the part of the equator joining the first two punctures. Then the involution T_{1} that reflects north-south across the equator takes the elements X = δ_{1}δ_{2}, Y = δ_{2}δ_{3}, and Z = δ_{3}δ_{1} of π_{1}N to δ_{1}^{−1}δ_{2}^{−1} = δ_{1}^{−1}X^{−1}δ_{1}, δ_{3}^{−1}δ_{2}^{−1} = Y^{−1}, and δ_{2}δ_{4}. The action on traces is (x, y, z) ↦ (x, y, xy − z + R). Similarly, by arraying the four punctures in the orders 1324 and 1243 around the equator we get two more involutions T_{2}, T_{3}; the actions on ℂ^{3} of all three involutions are the three involutions τ_{1}, τ_{2}, τ_{3} above.

It is not hard to see that the pairwise products T_{3}T_{1}, T_{1}T_{2}, and T_{2}T_{3} of these involutions are the Dehn twists T_{X}, T_{Y} , T_{Z}, on the three separating simple closed curves that represent the elements X, Y, Z of π_{1}N. These Dehn twists generate π_{0} Homeo^{+}(N, ∂_{1}N, …, ∂_{4}N) (in fact, any two of them do, and T_{X}T_{Y} T_{Z} = 1, see e.g., [1]). It follows that T_{1}, T_{2}, T_{3} generate π_{0} Homeo(N, ∂_{1}N, …, ∂_{4}N). Moreover, they generate it as a free product C_{2} ∗ C_{2} ∗ C_{2} since the action on ℂ^{3} is as this free product. Summarizing:

Theorem 5.3. The mapping class group π_{0} Homeo(N, ∂_{1}N, …, ∂_{4}N) is the free product C_{2} ∗ C_{2} ∗ C_{2} generated by the three involutions T_{1}, T_{2}, T_{3} above, and it acts on the relative character varieties via the action of C_{2} ∗ C_{2} ∗ C_{2} of the previous theorem. □

Note that our computation implies that the Dehn twists T_{X} and T_{Y} act on the relative character varieties by the following actions on ℂ^{3}

τ_{X}: (x, y, z) ↦ (x, x^{2}y − xz + Rx − y + Q, xy − z + R)
τ_{Y} : (x, y, z) ↦ (yz − x + P, y, y^{2}z − xy + P y − z + R) .

This agrees with the calculation in [7] (except for sign differences because we changed coordinates to reverse the signs of x, y, z). But it should not agree, since our Dehn twists are actually the inverses of the Dehn twists of [7]. The discrepancy is because automorphisms γ of π_{1}N act on characters χ by (γ, χ) ↦ γχ := χ ∘ γ^{−1}, but [7] uses (γ, χ) ↦ χ ∘ γ, which gives a right-action.

Theorem 5.4. There are natural geometric isomorphisms:

π_{0} Homeo(N, ∂_{1}N, ∂N − ∂_{1}N) ≅ PGL(2, ℤ)
π_{0} Homeo(N, ∂_{1}N, …, ∂_{4}N) ≅ PGL(2, ℤ)_{(2)}
π_{0} Homeo(N, ∂_{N}) ≅ PGL(2, ℤ)_{(2)} ⋊ S_{4}
≅ PGL(2, ℤ) ⋉ Σ = Γ

Proof. This is mostly well-known, so we describe it briefly. GL(2, ℤ) acts on T^{2} = ℝ^{2}/ℤ^{2} by linear maps, giving an isomorphism of GL(2, ℤ) with the mapping class group of T^{2}. The central element −I ∈ GL(2, ℤ) acts as an involution fixing the four half-integer points

(0, 0), (\frac{1}{2}, 0), (0, \frac{1}{2}), (\frac{1}{2}, \frac{1}{2}),

so the quotient PGL(2, ℤ) = SL(2, ℤ)/{±I} acts on the orbit space, which is S^{2}, and permutes the four branch points. This action fixes the first branch point and permutes the other three via the quotient PGL(ℤ/2, 2) (isomorphic to S_{3}). Thus the kernel

PGL(2, ℤ)_{(2)} = Ker(PGL(ℤ, 2) → PGL(ℤ/2, 2))

acts on the 2-sphere fixing all four branch-points. By replacing each branch point by the circle of tangent directions at the point, we get actions of PGL(2, ℤ) and PGL(2, ℤ)_{(2)} on the four-holed sphere N. Since a mapping class of the four-holed sphere gives a mapping class of the four-pointed sphere, which then lifts, up to C_{2} ambiguity, to a mapping class on the torus and thence to an action on its homology, this gives the first two isomorphisms of the theorem.

Finally, the mapping class group π_{0} Homeo(N, ∂N) acts on the set of boundary components, giving a homomorphism to S_{4}, determining it as an S_{4} extension of PGL(2, ℤ)_{(2)} = π_{0} Homeo(N, ∂_{1}N, …, ∂_{4}N):

1 → PGL(2, ℤ)_{(2)} → π_{0} Homeo(N, ∂N) → S_{4} → 1

This is a split extension, since we can map S_{4} to π_{0} Homeo(N, ∂N) as follows: Construct N from the boundary of a regular tetrahedron by removing neighborhoods of the vertices; the tetrahedral group S_{4} then acts by rigid motions on N. The Klein four-group Σ ⊂ S_{4} ⊂ π_{0} Homeo(N, ∂N) is the kernel of the action of π_{0} Homeo(N, ∂N) on homology of the torus T_{2} discussed above: each element of the Klein four-group comes from a translation of T^{2} = ℝ^{2}/ℤ^{2} by an element of the half-integer lattice, and these act trivially on homology. Summarizing, we see that π_{0} Homeo(N, ∂N) fits in a diagram

1  1
↓  ↓
Σ  ⟶^{=} Σ
↓  ↓
1 ⟶ PGL(2, ℤ)_{(2)} ⟶ π_{0} Homeo(N, ∂N) ⟶ S_{4} ⟶ 1
↓^{=}  ↓  ↓
1 ⟶ PGL(2, ℤ)_{(2)} ⟶  PGL(2, ℤ)  ⟶ S_{3} ⟶ 1
↓  ↓
1  1

This is like the one for Γ in the Introduction. In fact, both diagrams represent the group in the middle as the pullback for the diagram

S_{4}
↓
PGL(2, ℤ) ⟶ S_{3}

so they are isomorphic, and π_{0} Homeo(N, ∂N) ≅ Γ. □

By explicitly lifting the above generating involutions T_{1}, T_{2}, T_{3} to the torus and computing action on homology one finds that they correspond under this isomorphism to the elements

(\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) , (\begin{matrix}1 & 0 \\ 2 & −1 \end{matrix}) , (\begin{matrix}1 & 2 \\ 0 & −1 \end{matrix})

respectively. The three Dehn twists T_{X} = T_{3}T_{1}, T_{Y} = T_{1}T_{2}, T_{Z} = T_{2}T_{3}, therefore correspond to

(\begin{matrix}1 & 2 \\ 0 & 1 \end{matrix}) , (\begin{matrix}1 & 0 \\ −2 & 1 \end{matrix}) , (\begin{matrix}1 & −2 \\ 2 & −3 \end{matrix}) .

We now return to the topology of the fibers of the polynomials κ_{P,Q,R}.

Theorem 5.5. Each fiber V_{t} of κ_{P,Q,R} has at most isolated singularities. It has reduced homology only in dimension 2, and H_{2}(V_{t}) = H_{2}(V)/I(V_{t}), where V is a typical non-singular fiber and I(V_{t}) is the “subgroup of vanishing cycles” for the singularities of V_{t}, which is nontrivial if and only if V_{t} is singular. Moreover, H_{2}(V) ≅ ℤ^{5}, and it is the direct sum of the groups of vanishing cycles I(V_{t}), summed over the singular fibers of κ_{P,Q,R} (in particular, there are at most five singular fibers).

Proof. Since the gradient dκ_{P,Q,R} differs from the gradient dκ by the constant vector (P, Q, R), Lemma 1.1 applies to κ_{P,Q,R}, so κ_{P,Q,R} is tame in Broughton’s sense. Broughton [2] (see also [13]) proves that tameness implies the statements of the theorem, except, of course, for the fact that H_{2}(V) ≅ ℤ^{5}. We will see this below, where we show that each non-singular fiber V of κ_{P,Q,R} has the same topology as the non-singular fibers of κ. □

That κ_{P,Q,R} has at most five singular fibers is also easy to see by direct computation, and moreover, that for generic P, Q, R it has exactly five singular fibers, each with a single quadratic singularity. However, more complex singularities can occur. Benedetto and Goldman discuss this topology in [5], and describe, in particular, under what conditions a relative character variety for N can be a singular fiber of the relevant cubic κ_{P,Q,R}.

In any case, we restrict now to a nonsingular fiber V = κ_{P,Q,R}^{−1}(t) of some κ_{P,Q,R}. Denote by \overline{V} the closure of V in ℙ^{3}(ℂ). As discussed in sections 3 and 4 for the case P = Q = R = 0, \overline{V} is a nonsingular projective cubic which intersects the projective plane at infinity in the three lines x = 0, y = 0, z = 0, and these are three of the 27 lines on the cubic surface \overline{V}, the others arising through singular fibers of the projection of V to the three coordinate axes. The computations are essentially the same as the case P = Q = R = 0. Thus the topology of the nonsingular cubic V does not change as the parameters P, Q, R, t that define it change. The action of the automorphism group of this cubic on homology, which is always a subgroup of the group Γ, will therefore also not change, so it is as described in the Introduction.

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[4] , The action of the Torelli group on the homology of representation spaces is nontrivial, Topology 39 (2000), 851–871.

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[8] Action of the modular group on real SL(2)–characters of a one-holed torus, Topology and Geometry (to appear). 2

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[11] Magnus, W., Rings of Fricke characters and automorphism groups of free groups, Math. Zeit. 170 (1980), 91–103. 14

[12] Neumann, W.D., A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981), 299–343. 10

[13] and Norbury, P., Unfolding singularities at infinity, Math. Annalen 318 (2000), 149–180. 5, 19

[14] Nielsen, J., Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei Erzeugenden, Math. Ann. 71 (1918), 385–397. 2

Department of Mathematics, The University of Maryland, College Park, MD 20742

E-mail address: wmg@math.umd.edu

Department of Mathematics, Barnard College, Columbia University, New York, NY 10027

E-mail address: neumann@math.columbia.edu