MODULI SPACES

WILLIAM M. GOLDMAN

June 17, 1995

This article summarizes a series of talks presented at the workshop held at the Korea National University of Education at Ch’onju on July 11, 1994. The talks were divided into segments of 50, 40 and 20 minutes each. I decided to lecture on this material partially to provide background for Ronnie Lee’s later lectures at this conference. In particular I wanted to describe some of the symplectic geometry which Professor Lee uses in his discussion of the Casson invariant.

For a more detailed survey of this material see my article “Geometric structures and varieties of representations,” in Contemporary Mathematics 1990, published by the American Mathematical Society.

1. Formal properties and basic examples

The first lecture dealt with the formal properties of such moduli spaces and a few basic examples. Spaces of homomorphisms give rise to a way in which topology and geometry are wedded via algebra. Topology enters through a closed oriented surface S, or rather through its fundamental group π. Geometry enters through a classical homogeneous space X, or rather through its automorphism group G. The space Hom(π, G) of homomorphisms is a natural object closely related to geometric structures on S and flat connections on G-bundles on M.

This general construction admits rich formal symmetries: Aut(π) × Aut(G) on Hom(π, G) by

( Aut(π)× Aut(G))× Hom(π, G)⟶ Hom(π, G)
( ϕ,  h)  · ρ  ⟼ h ∘ ρ ∘ ϕ^{−1}

In practice we form the quotient by the inner automorphisms in Aut(G) to obtain a deformation space Hom(π, G)/G whose points parametrize flat principal G-bundles. The inner automorphisms of π act trivially on this quotient so the resulting action on Hom(π, G)/G factors through the action of the outer automorphism group Out(π).

A basic problem is to determine the geometric structure on Hom(π, G)/G which is invariant under Out(π).

The author acknowledges support from National Science Foundation grants DMS-8613576, DMS-8902619

1.1. The basic examples. When π is the fundamental group of a closed Riemann surface M, then Out(π) is the mapping class group π_{0}(Diff(M)). In this case different choices of Lie group G produce diverse behavior of moduli space geometry.

G = ℝ: Maybe the simplest Lie group is the additive group of real numbers ℝ. In this case the moduli space is H^{1}(M; ℝ). Its natural structure is that of a symplectic vector space under the symplectic form given by:

H^{1}(M; ℝ)×H^{1}(M; ℝ)⟶ℝ
(α,  β)  ⟼(α ∪ β) ∩ [M]

The mapping class group acts (non-effectively) via a homomorphism

Out(π) ⟶ Sp(ℤ^{2g}) ⊂ Sp(ℝ^{2g}).

When M is given a complex structure, then the moduli space inherits the structure of a Hermitian vector space: we can represent elements of H^{1}(M, ℝ) by harmonic 1-forms and the Hodge star operator (which takes a harmonic 1-form to its harmonic conjugate 1-form) is a complex structure on the space of harmonic 1-forms. However, this structure is not invariant under the mapping class group. G = U(1): When G is the circle group, then the above remarks apply as well. Now the moduli space identifies with the Jacobi variety of M, when M is given the structure of a Riemann surface. This space parametrizes topologically trivial holomorphic line bundles over M and is a Kähler manifold. Although the Kähler structure (in particular the complex structure) varies with the complex structure on M, the symplectic structure is a topological invariant of M. G = U(n): Study of the higher rank case was initiated by Narasimhan and Seshadri who showed the moduli space of flat U(n)-bundles over M is the moduli space parametrizing equivalence classes of semistable holomorphic ℂ^{n}-bundles over M. In particular they showed this moduli space has the structure of a complex projective variety, and this structure varies as one changes the complex structure on M. However, its symplectic structure is independent of the complex structure, and the mapping class group acts by symplectomorphisms on this moduli space. This space was reinterpreted by Atiyah and Bott as the moduli space of solutions of the Yang-Mills equations over a Riemann surface. Their work emphasizes the topological invariance of the symplectic structure. The mapping class group acts effectively on this space, preserving the symplectic structure (and hence a volume form). For G = SU(2) this action is ergodic. Perhaps the mapping class group acts ergodically on Hom(π, G)/G for any compact Lie group G,.

G = PSL(2, ℝ): Representations of the fundamental group arise from uniformizations of M — representations of M as quotients of the hyperbolic plane H^{2} by discrete groups. The relevant representations are embeddings of π onto discrete subgroups of the group G ≅ PSL(2, ℝ) of orientationpreserving isometries of H^{2}. The moduli space of such representations is the Teichmüller space 𝔗_{M} of M, a 6(g − 1)-dimensional cell constituting a connected component of Hom(π, G)/G. The corresponding symplectic structure on 𝔗_{M} is associated to the Weil-Petersson Kähler structure and is independent of the complex structure of M. The topological flavor of this symplectic structure is evident in Wolpert’s work on the symplectic geometry of 𝔗_{M}: Wolpert showed that the Fenchel-Nielsen twist flows on 𝔗_{M} were Hamiltonian (although they are not holomorphic). Furthermore the Fenchel-Nielsen coordinates on 𝔗_{M} are canonical coordinates (global Darboux coordinates) for 𝔗_{M} as a symplectic manifold. G = U(n, 1): In this case the space Hom(π, G)/G contains both

Hom(π, SU(1, 1))/ SU(1, 1) and Hom(π, SO(2, 1))/ SO(2, 1)

Both subspaces contain discrete embeddings. By recent joint work with Kapovich and Leeb, at least 2g − 1 components of Hom(π, G)/G contain discrete embeddings. G = SL(3, ℝ): In this case Hitchin showed that a connected component of Hom(π, G)/G is diffeomorphic to a cell of dimension 16g − 16. Joint work with Suhyoung Choi implies that this connected component consists of convex real projective structures, that is, uniformizations of M as quotients of convex domains in P^{2}(ℝ) by discrete groups of collineations. Hom(π, G)/G has at least 3 connected components. We conjecture that Hom(π, G)/G has exactly three components.

2. Hyperbolic structures on surfaces

The second lecture considered the case G = PSL(2, ℝ). Orientation-preserving discrete embeddings comprise one component of Hom(π, G)/G, which may be identified with the Teichmüller space 𝔗_{S} of S. This space has a Kähler geometry arising from a complex structure (discovered by Ahlfors and Teichmüller) and the Weil-Petersson metric. This Kähler structure is invariant under Out(π), which acts properly discontinuously. The quotient 𝔗_{S}/Out(π) is the Riemann moduli space which parametrizes conformal equivalence classes of Riemann surfaces of genus g.

An important integer-valued invariant, which distinguishes the connected components of Hom(π, G)/G is the Euler class, discussed in detail in my doctoral dissertation. Associated to a representation ρ : π ⟶ G is an oriented H^{2}-bundle H_{ρ}.

Oriented H^{2}-bundles are classified by the Euler class. The resulting map

e : Hom(π, G)/G ⟶ ℤ

expresses the topological type of a flat PSL(2, ℝ)-bundle over S. By a classical result of Milnor and Wood, the Euler number satisfies

|e(ρ)| ≤ |χ(S)| = 2g − 2

and every such integer occurs. Moreover the connected components are the 4g − 3 inverse images e^{−1}(j) for j = 2 − 2g, …, 0, …, 2g − 2.

These other components relate to moduli spaces of singular hyperbolic structures. Let S be a surface and let p_{1}, …, p_{k} ∈ S be points. Let θ_{i} > 0 for i = 1, …, k. We consider geometric structures on M which are locally modelled on H^{2} on S − {p_{1}, …, p_{k}} and such that each p_{i} has a coordinate neighborhood isometric to a hyperbolic cone of cone angle θ_{i}.

Let M be a Riemann surface and choose singular data (p_{1}, θ_{1}), …, (p_{k}, θ_{k}) ∈ M × ℝ_{+} as above. Then there exists a singular conformal structure on S which agrees with the original conformal structure on M − {p_{1}, …, p_{k}} and has conical singularities at p_{i} of cone angle θ_{i}. If

e(M; θ_{1}, …, θ_{k}) := χ(M) − ∑_{i=1}^{k} \frac{θ}{2π} − 1 < 0

then by a result of McOwen and Troyanov, there exists a singular hyperbolic structure on S with cone angles θ_{i} at each p_{i} in this singular conformal structure. Since the hyperbolic structure on M − {p_{1}, …, p_{k}} is nonsingular, there is a holonomy homomorphism π_{1}(M − {p_{1}, …, p_{k}}) ⟶ PSL(2, ℝ) sending a simple loop encircling p_{i} to an elliptic element of rotation angle θ_{i}. In particular if all the θ_{i} are multiples of 2π, this homomorphism extends to π_{1}(M) and has Euler number e(M; θ_{1}, …, θ_{k}).

For j = 1, …, 2g − 2, let Σ_{j} be the bundle over 𝔗_{M} whose fiber over the Riemann surface M is the j-th symmetric product Symm^{j}(M) of M; it is a complex manifold of complex dimension 3g − 3 + j which parametrizes singular conformal structures with cone angles θ_{i} = 2π(1 + m_{i}) at p_{i}, where the positive integer m_{i} is the multiplicity of p_{i} in the corresponding effective divisor in Symm^{j}(M). The above singular uniformization theorem gives a map

h : Σ_{j} ⟶ e^{−1}(2g − 2 + j)

where ∑_{i=1}^{k} m_{i} = j. It follows from the theory developed by Hitchin that this map is a homotopy- equivalence.

The uniformization map h is equivariant with respect to the action of the mapping class group, which acts properly on Σ^{j} (since it factors through the proper action on Teichmüller space). In contrast, the action on components other than Teichmüller space (e^{−1}(j) for |j| < 2g − 2) is more dynamically interesting. One may try to investigate this action by viewing h as a “resolution” of this action, studying instead the dynamical properties of the foliation induced by h on the quotient of Σ_{j}/ Out(π).

2.1. Examples of hyperbolic structures with conical singularities. Here are two constructions of singular hyperbolic structures on a closed oriented surface Σ with a single cone point. These constructions agree when Σ has genus two and the cone point has cone angle 4π.

The first construction uses a branched covering and identifies the holonomy representation as a non-injective homomorphism onto a discrete subgroup of PSL(2, ℝ). Consider the involution f : Σ ⟶ Σ with two fixed points whose quotient orbifold O is a torus T with two branch points p_{1}, p_{2} of order two. The fundamental group of O is

π_{1}(O) ≅ ⟨A, B, C_{1}, C_{2} | ABA^{−1}B^{−1}C_{1}C_{2} = C_{1}^{2} = C_{2}^{2} = I⟩

and χ(O) = 2 − 2g − ∑_{j}(1 − (n_{j})^{−1}) = −1. (Here the generator C_{i} corresponds to the loop which encircles the singular point p_{i}.) Corresponding to the quotient map q : Σ ⟶ O (a covering space of orbifolds) is an inclusion q_{∗} : π_{1}(Σ) ⟶ π_{1}(O) onto an index-two subgroup.

Consider a hyperbolic structure O′ on T with a cone point of angle π at p_{1} and no singularity at p_{2} (cone angle 2π). Then there is a homeomorphism h : O ⟶ O′ which induces a singular hyperbolic structure on O. Then

π_{1}(O′) ≅ ⟨A, B, C | ABA^{−1}B^{−1}C = C^{2} = I⟩ ⊂ PSL(2, ℝ)

and we arrange that h_{∗}(C_{1}) = C and the kernel of h_{∗} : π_{1}(O) ⟶ π_{1}(O′) is normally generated by C_{2}. We pull back the singular hyperbolic structure on O induced by h : O ⟶ O′ by the double covering q : Σ ⟶ O to obtain a singular hyperbolic structure on Σ with a cone angle of 2π at h^{p_{1}} (a smooth point) and cone angle of 4π at h^{p_{2}}. We see that for this hyperbolic structure, the Euler class is the negative of

area /2π = 2g − 2 + ∑_{j}(1 − α_{j}/2π) = 2(2) − 2 + (1 − 2) = 1

as expected. By varying the hyperbolic structure on the orbifold O′ we obtain a family of representations of π_{1}(Σ) with Euler class -1 parametrized by ℝ^{2}.

Here is another construction of singular hyperbolic structures on a surface of genus g with a single cone point for each Euler number 2 − 2g ≤ e < 0, which agrees with one of the above structures for g = 2, e = −1 . Begin by choosing an angle

0 < θ < \frac{4g − 2}{4g}π

(For an integer 2 − 2g ≤ j < 0,

θ = 2π \frac{j − 1 + 2g}{4g}

will produce an example on a surface of genus g with Euler number j.) Now let P = P (g, θ) be a regular 4g-gon with angle θ. (To construct P, choose a center O and choose rays l_{1}, …, l_{4g} at O at angles π/(2g) from each other. For each i = 1, …, 4g, let v_{i} be the point at distance

cosh^{−1} (cot(\frac{θ}{2}) cot(\frac{π}{4g}))

from O along l_{i}. Then v_{1}, …, v_{4g} are the vertices of P.)

For each i = 1, …, g there exist unique isometries A_{i}, B_{i} ∈ PSL(2, ℝ) such that A_{i} takes the edge with endpoints

v_{4g−4+i+3}, v_{4g−4+i+4}

to the edge with endpoints

v_{4g−4+i+2}, v_{4g−4+i+1}

and B_{i} takes the edge with endpoints

v_{4g−4+i+2}, v_{4g−4+i+3}

to the edge with endpoints

v_{4g−4+i+4}, v_{4g−4+i+5}

(where v_{j} = v_{k} for j ≡ k( mod 4g)). Then A_{1}, B_{1}, …, A_{g}, B_{g} satisfy the standard relation

A_{1}B_{1}A_{1}^{−1}B_{1}^{−1} … A_{g}B_{g}A_{g}^{−1}B_{g}^{−1} = I

and define identifications on ∂P such that the quotient Q = P/ ≈ is a singular hyperbolic structure. Each point in the interior of P, or in the interior of an edge of ∂P, is a smooth point of Q, but the point of Q corresponding to a vertex of P is a singularity of cone angle:

α = 4gθ

The Euler class of the holonomy representation is given by the area formula:

(2 − 2g) + (1 − \frac{α}{2π}) = j

where j = 2 − 2g, …, −2, −1. For j = 2 − 2g, the structure is nonsingular and corresponds to the Fuchsian representation.

3. The symplectic structure

In the last part of my talk, I discussed the general construction of the symplectic structure and gave an application to 3-manifolds. Let 𝔤 be the Lie algebra of G. Suppose there exists a nondegenerate symmetric bilinear form B : 𝔤 × 𝔤 ⟶ ℝ invariant under the adjoint representation. (For example if G is a reductive matrix group, such as any of the above examples, then B(X, Y) = tr(XY) is such a pairing.) If ρ ∈ Hom(π, G), then there is a flat vector bundle 𝔤_{Ad ρ} whose fiber is 𝔤. Then B defines a pairing of local systems 𝔤_{Ad ρ} × 𝔤_{Ad ρ} ⟶ ℝ

Recall that the “Zariski tangent space” to Hom(π, G)/G at an equivalence class [ρ] identifies with the cohomology group H^{1}(M; 𝔤_{Ad ρ}). The corresponding pairing

H^{1}(M; 𝔤_{Ad ρ})×H^{1}(M; 𝔤_{Ad ρ})⟶ℝ
(α,  β)  ⟼B_{∗}(α ∪ β) ∩ [M]

defines a nondegenerate alternating bilinear mapping ω_{[ρ]} on each tangent space T_{[ρ} Hom(π, G)/G. The corresponding exterior 2-form ω on Hom(π, G)/G is closed, and hence defines a symplectic structure on Hom(π, G)/G which includes all the symplectic structures mentioned above.

As a simple application of this construction, suppose that W^{3} is a compact 3manifold with boundary ∂W = S. The inclusion i : S ↪ W induces a map

i_{#} : Hom(π_{1}(W), G) ⟶ Hom(π, G)/G

which is isotropic, that is, (i_{#})^{∗}ω = 0. In other words, the representations which extend to W form an isotropic subvariety of Hom(π, G)/G. (In particular this subspace has dimension at most half that of Hom(π, G)/G.) To prove this assertion, let α, β ∈ H^{1}(W; 𝔤_{Ad ρ}). Then

ω_{[ρ]}((i_{#})_{∗}α), (i_{#})_{∗}β)) = B_{∗}((i_{#})_{∗}α), (i_{#})_{∗}β)) ∩ [M]
= B_{∗}(α ∪ β) ∩ i^{∗}[M] = 0

since i^{∗}[M] = ∂[W] is the zero cohomology class on W.

Particularly interesting is the case when W is a solid handlebody of genus g. Consider the case that G = SU(2). Then Hom(π, G)/G has dimension 6g − 6 and since π_{1}(W) is free of rank g − 1, Hom(π_{1}(W), G) ≅ G^{g−1} and Hom(π_{1}(W), G)/G has dimension 3g −3. Furthermore the mapping i_{#} : Hom(π_{1}(W), G)/G ⟶ Hom(π, G)/G is an embedding. It follows that its image is a Lagrangian subvariety of Hom(π, G)/G, a fact which has played an important role in the construction of recent 3-manifold invariants, such as those discussed in Ronnie Lee’s lectures at this conference.

Department of Mathematics University of Maryland, College Park, MD 20742 E-mail address: wmg@math.umd.edu