Notes on Geometry and 3-Manifolds, Study notes of Geometry

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Notes on Geometry and 3-Manifolds

Walter D. Neumann

Appendices by Paul Norbury

Preface

These are a slightly revised version of the course notes that were distributed during the course on Geometry of 3-Manifolds at the Tur´an Workshop on Low Dimensional Topology in Budapest, August 1998. The lectures and tutorials did not discuss everything in these notes. The notes were intended to provide also a quick summary of background material as well as additional material for “bedtime reading.” There are “exercises” scattered through the text, which are of very mixed difficulty. Some are questions that can be quickly answered. Some will need more thought and/or computation to complete. Paul Norbury also created problems for the tutorials, which are given in the appendices. There are thus many more problems than could be addressed during the course, and the expectation was that students would use them for self study and could ask about them also after the course was over. For simplicity in this course we will only consider orientable 3-manifolds. This is not a serious restriction since any non-orientable manifold can be double covered by an orientable one. In Chapter 1 we attempt to give a quick overview of many of the main concepts and ideas in the study of geometric structures on manifolds and orbifolds in dimen- sion 2 and 3. We shall fill in some “classical background” in Chapter 2. In Chapter 3 we then concentrate on hyperbolic manifolds, particularly arithmetic aspects.

Lecture Plan:

1. Geometric Structures.
2. Proof of JSJ decomposition.
3. Commensurability and Scissors congruence.
4. Arithmetic invariants of hyperbolic 3-manifolds.
5. Scissors congruence revisited: the Bloch group.

4 1. GEOMETRIC STRUCTURES

A manifold with geometric structure modeled on a geometry X is isometric to X/Γ for some discrete subgroup Γ of the isometry group of X. This can be proved by an “analytic continuation” argument (key words: developing map and holonomy, cf. e.g., [ 46 ]). We will not go into this, since for our purposes, we can take it as definition:

Definition 1.1. A geometric manifold (or manifold with geometric structure) is a manifold of finite volume of the form X/Γ, where X is a geometry and Γ a discrete subgroup of the isometry group Isom(X).

We will usually restrict to orientable manifolds and orbifolds for simplicity. That is, in the above definition, Γ ⊂ Isom+(X), the group of orientation preserving isometries.

2. Geometries in dimension 2. There are three basic geometries in dimension 2:

S^2 = {(x, y, z) ∈ R^3 | x^2 + y^2 + z^2 = 1}, ds =

dx^2 + dy^2 + dz^2 , Curvature K = 1; E^2 = R^2 with metric , ds =

dx^2 + dy^2 , Curvature K = 0; H^2 = {z = x + iy ∈ C | y > 0 },

ds =

y

dx^2 + dy^2 , Curvature K = −1;

called spherical, euclidean, and hyperbolic geometry respectively. We will explain “curvature” below. Historically, euclidean geometry is the “original” geometry. Dissatisfaction with the role of the parallel axiom in euclidean geometry led mathematicians of the 19th century to study geometries in which the parallel axiom was replaced by other versions, and hyperbolic geometry (also called “Lobachevski geometry”) and elliptic geometry resulted. Elliptic geometry is what you get if you identify antipodally opposite points in spherical geometry, that is, it is the geometry of real 2-dimensional projective space. It is not a geometry in our sense, because of our requirement of simply connected underlying space, while spherical geometry is not a geometry in the sense of Euclid’s axioms (with modified parallel axiom) since those axioms require that distinct lines meet in at most one point, and distinct lines in spherical geometry meet in two antipodally opposite points. But from the point of view of providing a “local model” for geometric structures, spherical and elliptic geometry are equally good. Our requirement of simple connectivity assures that we have a unique geometry with given local structure and serves some other technical purposes, but is not really essential.

2.1. Meaning of Curvature. If ∆ is a triangle (with geodesic—i.e., “straight- line”—sides) with angles α, β, γ then α + β + γ − π = Kvol(∆) where vol means 2-dimensional volume, i.e., area. Because of our homogeneity assumption, our 2-dimensional geometries have constant curvature K, but general riemannian ge- ometry allows manifolds with geometry of varying curvature, that is, K varies from point to point. In this case K(p) can be defined as the limit of (α+β+γ−π)/ vol(∆)

4. SOME EXAMPLES OF GEOMETRIC MANIFOLDS AND ORBIFOLDS 5

over smaller and smaller triangles ∆ containing the point p, and the above formula must be replaced by α + β + γ − π =

∆ K d(vol). You may already have noticed that we have not listed all possible 2-dimensional geometries above. For example, the 2-sphere of radius 2 with its natural metric is certainly a geometry in our sense, but has curvature K = 1/4, which is different from that of S^2. But it differs from S^2 just by scaling. In general, we will not wish to distinguish geometries that differ only by scalings of the metric. Up to scaling, the above three geometries are the only ones in dimension 2.

3. Orbifolds An n-orbifold is a space that looks locally like Rn/G where G is a finite subgroup of GL(n, R). Note that G varies from point to point, for example, a neighborhood of [x] ∈ Rn/G looks like Rn/Gx where Gx = {g ∈ G | gx = x}. We will restrict, for simplicity, to locally orientable 2-orbifolds (i.e., the above G preserves orientation). Then the only possible local structures are R^2 /Cp, p = 1 , 2 , 3 ,... , where Cp is the cyclic group of order p acting by rotations. the local structure is then a “cone point” with “cone angle 2π/p” (Fig. 1).

fundamental domain

Figure 1

4. Some examples of geometric manifolds and orbifolds Example 4.1. Let Λ be a group of translations of E^2 generated by two linearly independent translations e 1 and e 2. Then T 2 = E^2 /Λ = gives a Euclidean structure on T 2. We picture a fundamental domain and how the sides of the fundamental domain are identified in Fig. 2. Note that different choices of {e 1 , e 2 } will give different Euclidean structures on T 2.

Example 4.2. A classical example is the subgroup PSL(2, Z) ⊂ PSL(2, R) = Isom+(H^2 ), which acts on H^2 with fundamental domain pictured in Fig. 3. The picture also shows how PSL(2, Z) identifies edges of this fundamental domain. thus the quotient H^2 / PSL(2, Z) is an orbifold with a 2π/2-cone-point at P , a 2π/3-cone- point at Q, and a “cusp” at infinity (Fig. 4). It has finite 2-volume (area) as you can check by integrating the volume form (^) y^12 dxdy over the fundamental domain:

vol(H^2 / PSL(2, Z)) = 2π/6. We give a different proof of this in the next section.

5. VOLUME AND EULER CHARACTERISTIC 7
6. Volume and euler characteristic Denote by Fg the compact orientable surface of genus g (Fig. 5 shows g = 4). Suppose it has a geometric structure with (constant) curvature K.

Figure 5

Subdivide Fg into many small geodesic triangles ∆i , i = 1, 2 ,... , T.

Let the number of vertices of this triangulation be V , number of edges E. It is known (Euler theorem) that

(1) T − E + V = 2 − 2 g.

By 1.1, K vol(∆i) = αi + βi + γi − π, where αi, βi, γi are the angles of ∆i, so

K vol(Fg ) =

∑^ T

i=

(αi + βi + γi − π) =

∑^ T

i=

(αi + βi + γi) − πT

= 2πV − πT

since we are summing all angles in the triangulation and around any vertex they sum to 2π. Now 3T = 2E (a triangle has 3 edges, each of which is on two triangles), so

(3) T = 2E − 2 T

(2) and (3) imply K vol(Fg ) = 2πV − 2 πE + 2πT. By (1) this gives: K vol(Fg ) = 2 π(2 − 2 g). Now suppose that instead of Fg we have the surface of genus g with s orbifold points with cone angles 2π/p 1 ,... , 2 π/ps. Then in step (2) above we must replace s summands 2π by 2π/p 1 ,... , 2 π/ps, so we get instead:

K vol(F ) = 2π

2 − 2 g −

∑^ s

i=

pi

If we also have h cusps we must correct further by subtracting 2πh. Thus:

Theorem 5.1. If the surface F of genus g with h cusps and s orbifold points of cone-angles 2 π/p 1 ,... , 2 π/ps has a geometric structure with constant curvature K then K vol(F ) = 2πχ(F )

with

χ(F ) = 2 − 2 g − h −

∑^ s

i=

pi

In particular, the geometry is S^2 , E^2 , or H^2 according as χ(F ) > 0 , χ(F ) = 0, χ(F ) < 0. §

8 1. GEOMETRIC STRUCTURES

There is a converse (which we will not prove here, but it is not too hard): Theorem 5.2 (Geometrization Theorem for Orbifolds). Let F be the orbifold of Theorem 5.1. Then F has a geometric structure unless F is in the following list:

1. g = 0, h = 0, and either s = 1 or s = 2 and p 1 6 = p 2 ;
2. g = 0, h = 1 or 2 , s = 0;
3. g = 0, h = 1, s = 1;
4. g = 0, h = 1, s = 2, p 1 = p 2 = 2.

That is: F ∼= X/Γ where X = S^2 , E^2 , or H^2 , and Γ ⊂ Isom+(X) is a discrete subgroup acting so X/Γ has finite volume. (In each of cases 2–4 the orbifold does have an infinite-volume Euclidean structure, which is unique up to similarity. The orbifolds of case 1 are “bad orbifolds,” that is, orbifolds which have no covering by a manifold at all—see Sect. 14)

Exercise 1. Here is a hyperbolic structure on F 2. Start with a regular octagon in H^2 with angles 2 π/ 8. Why does it exist? (Hint: work in the Poincar´e disk model instead of upper half plane and expand a regular octahedron centered at the origin from very small to very large.) Identify edges as shown in Fig. 6.

Figure 6

Note that we don’t really need a regular octahedron for this construction. All we need is:

(i) sum of all eight angles is 2π. (ii) pairs of edges to be identified have equal length.

It is not hard to see that this gives 8 degrees of freedom for choosing the polygon. This gives 6 degrees of freedom for the geometric structure on F 2 since the choice of the point P in F 2 uses 2 degrees of freedom. In general, Theorem 5.3. The set of geometric structures on the orbifold of Theorem 5. (up to isometry) is a space of dimension max{ 0 , 6 g − 6+2s +2h}. It is the so-called Teichm¨uller moduli space of geometric structures on the orbifold. (In the E^2 case, we must take structures up to similarity instead of isometry.)

10 1. GEOMETRIC STRUCTURES

We can compute the size of G from theorem 5.1:

|G| = vol(S^2 )/ vol(F ) = 4π/

2 π

[

pi

)])

Note that G = π 1 (F ). With Theorem 14.3 this gives the standard presentation of the above finite G ⊂ SO(3).

8. Dimension 3. The geometrization conjecture We have seen that in dimension 2 essentially “all” orbifolds have geometric structures. Until 1976, the idea that anything similar might hold for dimension 3 was so absurd as to be unthinkable. In 1976 Thurston formulated his

Conjecture 8.1 (Geometrization Conjecture). Every 3-manifold has a “nat- ural decomposition” into geometric pieces.

The decomposition in question had already been proved in a topological version by Jaco & Shalen and Johannson, as we describe in Chapter 2. One may assume by earlier results of Knebusch and Milnor (cf. [ 27 ]) that M 3 is connected-sum-prime, and M 3 then has a natural “JSJ decomposition” (also called “toral decomposition”) which cuts M 3 along embedded tori^1 into pieces which are one of

1. Seifert fibered with circle fibers (Seifert fibered means fibered over an orbifold, see Chapter 2);
2. not Seifert fibered with circle fibers but Sefert fibered over a 1-orbifold (i.e., S^1 or I) with T 2 -fibers (in the I case there are special fibers—Klein bottles—over the endpoints of I);
3. simple (see below), but not Seifert fibered. We will discuss the geometrization conjecture for these three cases after defining “simple.” We are interested in a manifold M 3 that is the interior of a compact

manifold-with-boundary M

3 such that ∂M 3 is a (possibly empty) union of tori (briefly “M 3 has toral ends”). This is because M 3 may have resulted via the JSJ decomposition theorem by cutting a compact manifold along tori.

Definition 8.2. M 3 is simple if every essential embedded torus (that is, one that doesn’t bound a solid torus in M 3 ) is isotopic to a boundary component of M.

The geometrization conjecture is true (and easy) in cases 1 and 2 above. In case 3 it splits into two conjectures:

Conjecture 8.3. A 3-manifold with |π 1 (M 3 )| < ∞ is homeomorphic to S^3 /G for some finite subgroup G ⊂ Isom+(S^3 ).

Conjecture 8.4. A simple 3-manifold with |π 1 (M 3 )| = ∞ which is not Seifert fibered has a hyperbolic structure.

Conjecture 8.3 is equivalent to the combination of two old and famous unsolved conjectures:

Conjecture 8.5 (Poincar´e Conjecture). π 1 (M 3 ) = { 1 } ⇒ M 3 ∼= S^3. Conjecture 8.6 (Space-Form-Conjecture). A free action of a finite group on S^3 is equivalent to a linear action.

(^1) The geometric version of the decomposition uses both tori and Klein bottles, see Sect. 6 of Chapter 2.

10. THE HYPERBOLIC GEOMETRIZATION CONJECTURE 11
    9. Geometrization in the “easy” cases There are 8 geometries relevant to 3-manifolds. They are called S^3 , Nil, PSL, Sol, H^3 , S^3 × E^1 , E^3 , H^2 × E^1. A Seifert fibered manifold with circle fibers is fibered M 3 → F 2 over an orbifold F 2 of dimension 2. If M 3 is closed there are two invariants associated with this situation:

• The orbifold euler characteristic χ(F 2 ) (see Section 3 of this chapter).
• The euler number of the Seifert fibration e(M 3 → F 2 ) (see Section 4 of Chapter 2).

and the relevant geometry is determined by these as:

χ > 0 χ = 0 χ < 0 e 6 = 0 S^3 Nil PSL e = 0 S^2 × E^1 E^3 H^2 × E^1

If the Seifert fibered manifold M 3 is not closed then e(M 3 → F 2 ) is not well defined (it depends on a choice of “slopes” on the toral ends of M 3 ) so M 3 can have either a PSL or a H^2 × E^1 structure. There are three exceptional cases: D^2 × S^1 , T 2 × (0, 1) and a manifold 2-fold covered by the latter (interval bundle over Klein bottle) each have infinite volume complete E^3 structures but no finite volume geometric structure. For more details on the above see [ 30 ] or [ 41 ]. Case 2 of the “easy cases” is manifolds with Sol-structures. Only closed mani- folds occur for this geometry. This leaves only H^3 to discuss.

10. The hyperbolic geometrization conjecture There are several models for H^3. One of the often used ones is the upper half space model :

{(z, r) ∈ C × R | r > 0 } with metric ds =

dx^2 + dy^2 + dr^2 /r,

where z = x + iy. The orientation preserving isometry group is Isom+(H^3 ) = PSL(2, C). Recall Conjecture 8.4: Conjecture. If M is simple and not Seifert fibered and |π 1 (M )| = ∞ then M has a hyperbolic structure.

This had been proved by Thurston in the following situations

• M 3 is Haken (see Chapter 2 for the definition; this is true if for instance if |H 1 (M )| = ∞)
• M has a finite symmetry with positive dimensional fixed point set,

but not all details of the second case are published yet (see [ 46 ], [ 29 ], [ 8 ]). This conjecture and the special cases proven so far have had a major effect on 3-manifold theory, including helping toward the solutions of several old conjectures, e.g., the Smith Conjecture ([ 29 ]) and various conjectures about knots.

12. RIGIDITY IN DIMENSION 2: TRIANGLE ORBIFOLDS 13

Figure 9

B´

A´

C´

C

B

D´

D

A

Figure 10

Example 11.4. Consider a regular ideal octahedron with faces identified as in Fig. 10 (match A to A′, B to B′, etc.) Result: Complement of the Whitehead link (Fig. 11).

12. Rigidity in dimension 2: triangle orbifolds A triangle orbifold is a 2-dimensional orbifold with g = 0, s + h = 3. A triangle orbifold is obtained by gluing together two copies of a triangle of the appropriate geometry (this triangle has ideal vertices if h > 0). We speak of the (p 1 , p 2 , p 3 )- triangle if the angles of the triangle are π/p 1 , π/p 2 , π/p 3 with pi ∈ { 2 , 3 , 4 ,... , ∞}. For example H^2 / PSL(2, Z) is the (2, 3 , ∞)-triangle orbifold.

14 1. GEOMETRIC STRUCTURES

Figure 11

A triangle orbifold has unique geometric structure, as does the orbifold S^2 /Cp with g = 0, h = 0, s = 2. In all other cases the dimension 6g − 6 + 2s + 2h of Teichm¨uller space is positive, so there are infinitely many geometric structures. Dimension 3 is in sharp contrast to this:

13. Mostow-Prasad rigidity Theorem 13.1 (Mostow-Prasad). The hyperbolic structure on a hyperbolic 3- orbifold is unique. In fact (stronger formulation): If Γ 1 and Γ 2 are discrete sub- groups of PSL(2, C) = Isom+(H^3 ) such that

(i) H^3 /Γ 1 is a finite volume orbifold, and (ii) Γ 1 ∼= Γ 2 ,

then any isomorphism Γ 1 → Γ 2 is induced by an inner automorphism (conjugation) in PSL(2, C). In particular, H^3 /Γ 1 ∼= H^3 /Γ 2 (isometry).

This is a remarkable result. The geometrization conjecture says that “almost every” 3-manifold has a hyperbolic structure, and rigidity says this structure is unique! Thus any information we extract from the geometry is actually a topological invariant of the manifold. Usually it is hard to find topological descriptions of the resulting invariants, but there is an elegant topological invariant of a 3-manifold called “Gromov norm” (after its inventor) which equals a constant multiple of the volume for a hyperbolic 3-manifold. In later lectures we will describe arithmetic invariants of the hyperbolic struc- ture. Again, by rigidity, these invariants are topological invariants.

14. Fundamental Group and Covering Spaces We have referred to fundamental groups and covering spaces of orbifolds above. For our purposes we can define the fundamental group of an orbifold similarly to the standard definition for spaces (our definition depends on the fact that we only consider orientable orbifolds—the codimension 1 sets of orbifold points that occur in non-orientable ones need extra technicalities): Let F be an orientable orbifold, ∗ ∈ F a base-point. π 1 (F ) = set of “homotopy classes” of closed paths γ : [0, 1] → F , γ{ 0 , 1 } = {∗} which do not pass through any orbifold points. “Homotopy” now means deformation of paths in which the deformation may pass through orbifold points, but when it does, the deformation in the “local picture” U/G must be the image of a deformation in U , cf. Fig. 12 (a 2 π 2 -cone-point).

16 1. GEOMETRIC STRUCTURES

Theorem 14.3. The orbifold F (g; h; p 1 ,... , ps) of genus g with h punctures and s orbifold points of the types p 1 ,... , ps has

π 1 (F (g; h; p 1 ,... , ps)) =〈a 1 ,... , ag , b 1 ,... , bg , q 1 ,... qs+h | qp 11 = 1,... , qp s s= 1, q 1... qs+h[a 1 , b 1 ]... [ag , bg ] = 1〉 .

15. Cusps of hyperbolic 3-orbifolds Where a hyperbolic 3-manifold is non-compact, the local structures is that of a cusp

C = {(x + iy, r) | r > K}/Λ

where Λ is a lattice of horizontal translations. Horizontal cross-sections (r = con- stant) of C are Euclidean tori which are shrinking in size as r → ∞.

Figure 13

The (r = constant) cross-sections of C are called horosphere sections. In a hyperbolic 3-orbifold the picture is the same except that the horosphere sections of a cusp can be any Euclidean orbifold. The Euclidean orbifolds are easily classified.

Exercise 2. Do this analogously to Section 7—solve the equation χ = 0 to show they are

• the Euclidean tori E^2 /Λ;
• the triangle orbifolds (cf. Section 12) with (p 1 , p 2 , p 3 ) = (2, 4 , 4), (2, 3 , 6), or (3, 3 , 3);
• orbifolds with g = 0, h = 0, s = 4, and (p 1 , p 2 , p 3 , p 4 ) = (2, 2 , 2 , 2). We call these “pillow orbifolds.” Exercise 3. Fig. 14 shows a pillow orbifold with an embedding of it as a the surface of a tetrahedron in E^3. Show that if one allows degenerate (flat) tetra- hedra, every pillow orbifold has a unique embedding as the surface of a euclidean tetrahedron up to isometry.

16. VOLUMES OF HYPERBOLIC ORBIFOLDS 17

Figure 14

The Euclidean triangle orbifolds are rigid (see Section 12) but the Euclidean tori and pillow orbifolds have 2-dimensional spaces of deformations of the Euclidean structure. In fact, every Euclidean torus double covers a unique Euclidean pillow orbifold and vice versa (the torus E^2 /Λ double covers E^2 /Γ, where Γ ⊂ Isom+(E^2 ) is generated by Λ and the map

0 − 1

: E^2 → E^2 ), so the Teichm¨uller moduli space of pillow orbifolds is the same as for tori, namely H^2 / PSL(2, Z)—see Theorem 6.

Definition 15.1. If a cusp of a hyperbolic 3-orbifold has horosphere sections which are tori or pillow orbifolds the cusp is called non-rigid, if the horosphere sections are triangle orbifolds the cusp is rigid.

The non-rigid cusps are important for “hyperbolic Dehn surgery,” as we shall describe later. One effect of this is that they affect the volume of the orbifold in a way that we can already describe.

16. Volumes of hyperbolic orbifolds Theorem 16.1. (Thurston, Jørgenson) The set Vol = {v ∈ R | v is the volume of some hyperbolic 3-orbifold} is a well ordered closed subset of R of order type ωω^. To each v ∈ Vol there are just finitely many orbifolds of this volume.

Otherwise expressed: the elements of Vol are ordered v 0 < v 1 < v 2 < · · · < vω < vω+1 < · · · < v 2 ω < · · · < v 3 ω < · · · · · · < vω^2 < · · · < vκ < · · ·.

The general index κ is an infinite ordinal number

κ = anωn^ + an− 1 ωn−^1 + · · · + a 0 and ai ∈ { 0 , 1 , 2 ,... }

If κ is divisible by ω then vκ is the limit of the vλ, λ < κ; we say vκ is a limit volume. If κ is divisible by ω^2 then vκ is a limit of limit volumes—it is a 2-fold limit volume. If κ is divisible by ωn^ then vκ is an n-fold limit volume (limit of (n − 1)-fold limit volumes).

Theorem 16.2. If M is a hyperbolic orbifold with n non-rigid cusps, then vol(M ) is an n-fold limit volume.

A few of the vκ are known: Colin Adams has found vω , v 2 ω , and v 3 ω. He has also found the six non-compact orbifolds of least volume (with rigid cusps; the smallest of these has been earlier found by Meyerhoff). See [ 1 ] and [ 2 ]. No-one knows v 0 (although there is a guess, namely 0. 03905.. ., known to be the smallest volume in the arithmetic case, [ 6 ]). The smallest hyperbolic manifold is even harder to determine, but again there is a guess, with volume about. 942707.. ., again known to be smallest among arithmetic hyperbolic 3-manifolds [ 7 ]. The minimal examples found so far are all arithmetic (see Chapter 3 or [ 33 ] for a definition). This is striking, because arithmetic examples are very sparse

19. NON-ORIENTABLE 2-ORBIFOLDS 19

Proposition 17.2. For any bound V there is a finite collection of hyperbolic orbifolds such that every hyperbolic orbifold with volume ≤ V results from Dehn surgery on a member of the collection.

18. Postscript If one believes the geometrization conjecture, and most topologists do, then understanding hyperbolic 3-manifolds is by far the biggest part of understanding all 3-manifolds. Many general topological questions remain unsolved for hyperbolic manifolds, for instance, does every hyperbolic manifold have a finite covering space with infinite homology, or which even fibers over S^1 (Thurston’s question)? This is not even known for the much more restricted class of arithmetic hyperbolic 3- manifolds. Of course, one of the most basic questions in most fields is the classification question. Can one find a reasonable classification of hyperbolic 3-manifolds? Al- though we are far from such a thing at present, it may not be an entirely hopeless project in the long run—Proposition 17.2 could be a starting point. An excellent tool for the computational study of hyperbolic 3-manifolds exists in Jeff Weeks’ computer program “snappea,” available for various computers, and the program “snap” based on it. These have already been used to help classify non-compact hy- perbolic manifolds that can be triangulated with few ideal simplices, for example.
19. Non-orientable 2-orbifolds If a 3-manifold is Seifert fibred over a 2-orbifold the 2-orbifold may be non- orientable even if the 3-manifold is orientable. The 2-orbifold will, however, be locally orientable. We will use g < 0 for genus of non-orientable surfaces (so g = − 1 , − 2 ,... is projective plane, Klein bottle, ...). The euler characteristic of the closed surface of genus g < 0 is 2 + g, so in theorem 5.1 we must replace 2 − 2 g by 2 + g if g < 0. Theorem 5.2 remains true with the additional exception: 2 ′. g = 0, h = 1, s = 0.

If one drops the condition of local orientability then one also has the local structures given by the dihedral groups of order 2n, n = 1, 2 ,....

Exercise 4. Generalise Theorems 5.1 and 5.2 to allow non-locally-orientable 2-orbifolds and then use the method of Section 7 to classify all compact spherical and euclidean orbifolds. This gives the classifications of all finite subgroups of O(3) and of the so called “seventeen wallpaper groups” (a misnomer, since most of the seventeen have positive dimensional Teichm¨uller spaces and are thus infinite families of groups).