3-manifolds

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1 Introduction

In the 3-dimensional setting there is no distinction between smooth, PL and topological manifolds neccesary; the categories of smooth, PL and topological manifolds are equivalent (TODO ref). A lot of techniques have been developed in the last century to study 3-manifolds but most of them are very special and don't generalise to higher dimensions. One key idea is to decompose manifolds along incompressible surfaces into smaller pieces, to which certain geometric models apply. A great progress was made in with the proof of the Poincaré conjecture and Thurton's geometrization conjecture by Perelman in 2003.

The universal cover of the famous Poincaré homology sphere is $S^3$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}S^3$ - here a view of the induced tesselation

2 Construction and examples

Basic examples are $\mathbb{R}^3, S^3, S^1 \times S$$\mathbb{R}^3, S^3, S^1 \times S$ with $S$$S$ any surface. Important types of 3-manifolds are Haken-Manifolds, Seifert-Manifolds, 3-dimensional lens spaces, Torus-bundles and Torus semi-bundles.

There are two topological processes to join 3-manifolds to get a new one. The first is the connected sum of two manifolds $M_1$$M_1$ and $M_2$$M_2$. Choose embeddings $f_1:D^3\rightarrow M_1$$f_1:D^3\rightarrow M_1$ and $f_2:D^3\rightarrow M_2$$f_2:D^3\rightarrow M_2$, remove the interior of $f_1(D^3)$$f_1(D^3)$ and $f_2(D^3)$$f_2(D^3)$ and glue $M_1$$M_1$ and $M_2$$M_2$ together along the boundaries $f_1(S^3)$$f_1(S^3)$ and $f_2(S^3)$$f_2(S^3)$. The second uses incompressible surfaces. Let $M$$M$ be manifold and $S\subset M$$S\subset M$ a surface. $S$$S$ is incompressible, if there is no disk $D$$D$ in $M$$M$ with $D\cap S=\partial D$$D\cap S=\partial D$. The torus sum is the process which glues incompressible tori boundary components together.

(TODO What is incompressibility needed? / What is is good for/ What happen if one takes a compressible surface ?)

3 Invariants

In the 3-dimensional world the fundamental group is a powerful invariant to distiguish manifolds. It determines already all homology groups:

• $H_1(M)$$H_1(M)$ = abelization of $\pi_1(M)$$\pi_1(M)$.
• $H_2(M) = H^1(M) = H_1(M)/$$H_2(M) = H^1(M) = H_1(M)/$torsion
• $H_3(M) = \Zz$$H_3(M) = \Zz$
• $H_n(M) = 0$$H_n(M) = 0$ for $n > 3$$n > 3$

4 Classification/Characterization

By reversing the process of connected and torus sum every 3-manifold can be decomposed into pieces which admit a geometric structure. We describe the details in the following.

4.1 Prime decomposition

Definition 4.1. A manifold $M$$M$ is called prime, if it can't be written as a non-trivial connected sum, i.e. $M=M_1 \# M_2$$M=M_1 \# M_2$ implies $M_1 = S^3$$M_1 = S^3$ or $M_2 = S^3$$M_2 = S^3$. A manifold $M$$M$ is called irreducible if every embedded $S^2$$S^2$ bounds a ball, i.e. the embedding extends to an embedding of $D^3$$D^3$.

Irreducibility is only slightly stronger than being prime. A orientable prime 3-manifold is either $S^2 \times S^1$$S^2 \times S^1$ or every embedded 2-sphere bounds a ball [Hempel1976, 3.8].

Theorem 4.2 Kneser. Every orientable, compact 3-manifold $M$$M$ has a decomposition $M=P_1 \# \ldots \# P_n$$M=P_1 \# \ldots \# P_n$ into prime manifolds $P_i$$P_i$ unique up to ordering and $S^3$$S^3$ summands.

Van Kampen's theorem tells you that $\pi_1(M \# N)=\pi_1(M)*\pi_1(N)$$\pi_1(M \# N)=\pi_1(M)*\pi_1(N)$. Hence any 3-manifold whose fundamental group cannot be written as a free product of two nontrivial subgroups can only be written as the connected sum of another 3-manifold with a simply connected 3-manifold. By the Poincaré conjecture a simply connected 3-manifold is already homeomorphic to $S^3$$S^3$. Hence each such manifold is prime.

Prime 3-manifolds can be distinguished by their fundamental groups into the following 3 types:

4.1.1 Type I: finite fundamental group

The universal cover $\tilde{M}$$\tilde{M}$ is a simply-connected 3-manifold. As the fundamental group already determines the homology of a oriented, closed compact 3-manifold, it has to be a homology sphere. Using the Hurewicz-theorem, its fundamental class is represented by a degree 1 map $S^3 \rightarrow \tilde{M}$$S^3 \rightarrow \tilde{M}$. This map induces isomorphisms on the homology and on the fundamental group. Hence it is a weak homotopy equivalence, and hence a homotopy equivalence by Whitehead's theorem (ref?). Hence every prime $3$$3$-manifold with finite fundamental group arises as the quotient of a homotopy sphere by a free action of a finite group. With the use of the Poincaré conjecture every homotopy 3-sphere is homeomorphic to $S^3$$S^3$ and we can write $M=S^3/\Gamma$$M=S^3/\Gamma$. If $\Gamma$$\Gamma$ is cyclic $M$$M$ is known as lens space (ref).

4.1.2 Type II: infinite cyclic fundamental group

By [Hempel1972, 5.2], $M$$M$ is a 2-sphere bundle over $S^1$$S^1$, that is, $S^2\times S^1$$S^2\times S^1$.

4.1.3 Type III: infinite non-cyclic fundamental group

Such a manifold $M$$M$ is always aspherical (TODO ref). The sphere theorem states, that every map $S^2\rightarrow M$$S^2\rightarrow M$ is homotopic to an embedding; and - as $M$$M$ is irreducible - it is nullhomotopic. Hence $\pi_2(M)=0$$\pi_2(M)=0$. Consider the universal covering $\tilde{M}$$\tilde{M}$ of $M$$M$. Its first homology vanishes as it is simply connected. The long exact sequence of homotopy groups of the fibration $\pi_1(M)\rightarrow \tilde{M}\rightarrow M$$\pi_1(M)\rightarrow \tilde{M}\rightarrow M$ gives a isomorphism $\pi_2(M)\cong \pi_2(\tilde{M})$$\pi_2(M)\cong \pi_2(\tilde{M})$. Hence by Hurewicz' theorem $H_2(\tilde{M})=0$$H_2(\tilde{M})=0$. Furthermore $H_3(\tilde{M})=0$$H_3(\tilde{M})=0$, as $M$$M$ is noncompact. Applying Hurewicz theorem again we get that all homotopy groups of $\tilde{M}$$\tilde{M}$ vanish and hence by Whitehead's theorem $\tilde{M}$$\tilde{M}$ is contractible. This means that $M$$M$ is apherical. Hence the homotopy type of a prime 3-manifold with infinite non-cyclic fundamental group is uniquely determined by its fundamental group. Furthermore not every group can occur as a fundamental group of a prime 3-manifold. The equivariant cellular chain complex of $\tilde{M}$$\tilde{M}$ is a projective resolution of the trivial $\Zz[\pi_1(M)]$$\Zz[\pi_1(M)]$-module $\Zz$$\Zz$. Hence .... For any subgroup $F\le \pi_1(M)$$F\le \pi_1(M)$ the space $\tilde{M}/F$$\tilde{M}/F$ is a finite-dimensional model for $K(F,1)$$K(F,1)$. For example a finite group cannot have such a model (by group homology ref) and hence $\pi_1(M)$$\pi_1(M)$ must be torsionfree. Furthermore it is a Poincaré duality group (link).

4.2 Torus decomposition

According to the previous section it remains to classify irreducible prime 3-manifolds. After cutting along spheres which don't bound balls as far as possible the next canonical step is to consider incompressible tori which are disjoint from the boundary.

Theorem 4.3 Jacob-Shalen, Johannson. If $M$$M$ is an irreducible compact orientable manifold, then there is a collection of disjoint incompressible tori $T_1, \ldots ,T_n$$T_1, \ldots ,T_n$ in $M$$M$ such that splitting $M$$M$ along the union of these tori produces manifolds $M_i$$M_i$ which are either Seifert-fibered or atoroidal, i.e. every incompressible torus in $M_i$$M_i$ is isotopic to a torus component of $\partial M_i$$\partial M_i$. Furthermore, a minimal such collection of tori $T_j$$T_j$ is unique up to isotopy in $M$$M$.

4.3 Geometrization

Thurston's geometrization conjecture (proven by Perelman) roughly states that all the pieces we get by this JSJ-decomposition admit one of eight possible geometric structures: There is a list of eight simply connected Riemannian manifolds - the so called model geometries. There is a subtlety that arises when trying to directly use the JSJ-decomposition, which is that we may come across a torus bundle over a circle, and the JSJ-decomposition would have us cut this into a torus bundle over a line, which does not admit a geometric structure, while the bundle over the line did. Once we modify the JSJ-decomposition to account for this, the pieces have a geometric structure.

A geometric structure on $M$$M$ is the choice of a complete homogeneous Riemannian metric on $M$$M$, with the property that its universal covering $\tilde{M}$$\tilde{M}$ equipped with the pull-back metric is isometric to one of the eight model geometries.

The eight model geometries are denoted $S^3$$S^3$, $E^3$$E^3$, $H^3$$H^3$, $S^2\times\mathbb R$$S^2\times\mathbb R$, $H^2\times\mathbb R$$H^2\times\mathbb R$, $\widetilde{\SL_2(\mathbb R)}$$\widetilde{\SL_2(\mathbb R)}$,

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$\rm Nil$, and

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$\rm Sol$. The first three are the three geometries with constant sectional curvature (i.e., they are isotropic), and the next two are products of isotropic two dimensional geometries with a line. The last three are Lie groups endowed with their unique left-invariant metric.

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$\rm Nil$ is the Heisenberg group, and

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$\rm Sol$ is a semi-direct product $\mathbb R^2\rtimes\mathbb R$$\mathbb R^2\rtimes\mathbb R$ with action $t(x,y)=(e^tx,e^{-t}y)$$t(x,y)=(e^tx,e^{-t}y)$.

The compact manifolds modeled on six of these geometries are all Seifert-fibered (and, conversely, all Seifert-fibered manifolds have such a geometry.) Specifically, in terms of the Euler characteristic $\chi$$\chi$ of the base orbifold and the euler number $e$$e$ of the bundle, we have the following classification of geometric structures on Seifert-fibered manifolds:

	χ>0	χ=0	χ<0
e=0	S^2×R	E^3	H^2×R
e≠0	S^3	Nil	~SL_2

The Seifert-fibered manifolds are well understood since the work of Seifert in the 30s (TODO: mention classification theorem).

Manifolds with the

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$\rm Sol$ geometry are finitely covered by torus bundles over $S^1$$S^1$ which are constructed via a hyperbolic gluing map (i.e., an automorphism of the torus whose eigenvalues don't have absolute value $1$$1$.) In fact, the covering is of degree at most $4$$4$, so the classification of these manifolds essentially reduces to the classification of torus bundles over $S^1$$S^1$.

The hyperbolic manifolds are the most mysterious. It follows from geometrization that the irreducible atoroidal 3-manifolds $M$$M$ such that $\pi_1(M)$$\pi_1(M)$ does not contain a subgroup isomorphic to $\mathbb Z\oplus\mathbb Z$$\mathbb Z\oplus\mathbb Z$ are exactly the hyperbolic 3-manifolds. This is because the other geometries are all either Seifert-fibered or torus bundles over the circle and so their fundamental groups necessarily contain a $\mathbb Z\oplus\mathbb Z$$\mathbb Z\oplus\mathbb Z$ subgroup. Conversely, the fundamental group of a hyperbolic 3-manifold cannot contain a $\mathbb Z\oplus\mathbb Z$$\mathbb Z\oplus\mathbb Z$ subgroup [Scott1983, Corollary 4.6].

Finite volume hyperbolic 3-manifolds are determined up to homeomorphism by their fundamental group by the Mostow-Prasad rigidity theorem (find reference), and such a manifold has a unique hyperbolic structure by the same theorem. So the classification of finite volume hyperbolic 3-manifolds is equivalent to the classification of finite covolume discrete subgroups of $SL_2(\mathbb C)$$SL_2(\mathbb C)$.

4.4 Dehn surgery

Dehn surgery is a way of constructing (TODO oriented ? neccesary) 3-manifolds. Given a link in a $3$$3$-manifold $N$$N$

$\displaystyle L: \coprod_{i=1}^n S^1\rightarrow N,$ $$\displaystyle L: \coprod_{i=1}^n S^1\rightarrow N,$$

and a choice of a tubular neighborhood of $L$$L$

$\displaystyle L': \coprod_{i=1}^n S^1\times D^2\rightarrow N\mbox{ with }L'(x,0)=L(x)$ $$\displaystyle L': \coprod_{i=1}^n S^1\times D^2\rightarrow N\mbox{ with }L'(x,0)=L(x)$$

.

(This choice essentially is the choice of a trivialization of the normal bundle; TODO find a correct formulation for this). This gives us a family of embedded, disjoint, full tori. The idea of Dehn surgery is to remove these Tori and glue them back in using a twist.\\ Let us restrict to the case with only one solid torus $L':S^1\times D^2\rightarrow N$$L':S^1\times D^2\rightarrow N$. Choose any self-homeomorphism $f$$f$ of the torus $S^1\times S^1$$S^1\times S^1$. The result of the Dehn surgery at $L$$L$ with the twist $f$$f$ is defined as

$\displaystyle N_{f,L'}:=N\setminus L'(S^1\times \mathring{D}^2) \cup_f S^1\times D^2=N\setminus L'(S^1\times \mathring{D}^2) \amalg S^1\times D^2/\sim,$ $$\displaystyle N_{f,L'}:=N\setminus L'(S^1\times \mathring{D}^2) \cup_f S^1\times D^2=N\setminus L'(S^1\times \mathring{D}^2) \amalg S^1\times D^2/\sim,$$

where the equivalence relation identifies for $(x,y)\in S^1\times S^1$$(x,y)\in S^1\times S^1$ the points $L'(x,y)$$L'(x,y)$ in the left component and $f(x,y)$$f(x,y)$ in the right component. If $f$$f$ is the coordinate flipping, Dehn surgery is nothing but usual codimension $2$$2$ surgery.

Lemma 4.4. Suppose $f,f'\in \Homeo(T^2)$$f,f'\in \Homeo(T^2)$ are isotopic and let $L':S^1\times D^2 \rightarrow N$$L':S^1\times D^2 \rightarrow N$ be any embedding of the full Torus in a $3$$3$-Manifold $N$$N$.Then $N_{f,L'}$$N_{f,L'}$ and $N_{f',L'}$$N_{f',L'}$ are homeomorphic.

Proof. Let $j:T^2\times [0;1] \rightarrow T^2$$j:T^2\times [0;1] \rightarrow T^2$ be an isotopy from $f$$f$ to $f'$$f'$. This gives a homeomorphism:

$\displaystyle J:T^2\times [0;1]\rightarrow T^2\times [0;1] \qquad (x,y,t)\mapsto (j(x,y,t),t).$ $$\displaystyle J:T^2\times [0;1]\rightarrow T^2\times [0;1] \qquad (x,y,t)\mapsto (j(x,y,t),t).$$

(TODO: is its inverse $(x,y,t)\mapsto (j(-,-,t)^{-1}(x,y),t)$$(x,y,t)\mapsto (j(-,-,t)^{-1}(x,y),t)$ continuous ?). The idea is to grab some additional space, where one can use the map $J$$J$. TODO

$\square$$\square$

TODO formulate a lemma, that M_{f,L'} also only depends on the isotopy class of $L'$$L'$ (which is hopefully true). Hence, we have to classify all self-homeomorphisms of $T^2$$T^2$ up to isotopy.

Lemma 4.5. Every self-homeomorphism of $T^2$$T^2$ is isotopic to exactly one homeomorphism of the shape

$\displaystyle f_A:\Rr^2/\Zz^2 \rightarrow \Rr^2 /\Zz^2 \qquad \left(\begin{array}{c}x\\y\end{array}\right)\mapsto A\cdot\left(\begin{array}{c}x\\y\end{array}\right),$ $$\displaystyle f_A:\Rr^2/\Zz^2 \rightarrow \Rr^2 /\Zz^2 \qquad \left(\begin{array}{c}x\\y\end{array}\right)\mapsto A\cdot\left(\begin{array}{c}x\\y\end{array}\right),$$

where $A\in GL_2(\Zz)$$A\in GL_2(\Zz)$ (reference of proof).

Proof. Since the torus is a $K\left(\Zz^2,1\right)$$K\left(\Zz^2,1\right)$-space, we have that $\pi_0 Map\left(T^2,T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$$\pi_0 Map\left(T^2,T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$ is an isomorphism. Homotopic surface homeomorphisms are isotopic (Reference?). Thus the restriction $\pi_0Homeo\left(T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$$\pi_0Homeo\left(T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$ is injective. Moreover, each $A\in Hom\left(\Zz^2,\Zz^2\right)$$A\in Hom\left(\Zz^2,\Zz^2\right)$ is realised by $f_A$$f_A$, therefore $\pi_0Homeo\left(T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$$\pi_0Homeo\left(T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$ is also surjective. TODO find a reference in ANY source about the mapping class group

$\square$$\square$

The next lemma tells us, that composition of self-homeomorphisms corresponds to two successive Dehn surgeries.

Lemma 4.6. Let $f,g\in \Homeo(T^2)$$f,g\in \Homeo(T^2)$ be given and let $L':S^1\times D^2\rightarrow N$$L':S^1\times D^2\rightarrow N$ is an embedding of the full torus in a 3-manifold. Then we have map

$\displaystyle L'':S^1 \times D^2 \rightarrow N\setminus L'(S^1\times \mathring{D}^2) \cup_f S^1\times D^2=N_{f,L'}$ $$\displaystyle L'':S^1 \times D^2 \rightarrow N\setminus L'(S^1\times \mathring{D}^2) \cup_f S^1\times D^2=N_{f,L'}$$

given by the map $S^1\times D^2\rightarrow S^1 \times D^2 \quad (x,y)(x,y/2)$$S^1\times D^2\rightarrow S^1 \times D^2 \quad (x,y)(x,y/2)$ postcomposed with the canonical inclusion in the second coordinate. Then $(N_{f,L'})_{g,L''} \cong N_{f\circ g,L'}$$(N_{f,L'})_{g,L''} \cong N_{f\circ g,L'}$. TODO right order of composition ? We will see in the proof.

Proof.

$\square$$\square$

We have to find out, which self-homeomorphisms of the torus don't change the homeomorphism type of the manifold.

Lemma 4.7. Consider a matrix of the form $\left( \begin{array}{cc} 1 & 0 \\ k & 1 \end{array}\right)$$\left( \begin{array}{cc} 1 & 0 \\ k & 1 \end{array}\right)$ and let $L':S^1\times D^2 \rightarrow N$$L':S^1\times D^2 \rightarrow N$ be any eembedding. Then $N_{f_A,L'}\cong N$$N_{f_A,L'}\cong N$.

TODO are there any orientation reversing homeos, that also extend ? Think so. Also add them here.

Proof. The homeomorphism $f_A \in \Homeo(T^2)$$f_A \in \Homeo(T^2)$ extends to a homeomorphism of $\bar{f_A}\in \Homeo(S^1\times D^2)$$\bar{f_A}\in \Homeo(S^1\times D^2)$:

$\displaystyle \bar{f_A}(x,y):=(x,x^ky),$ $$\displaystyle \bar{f_A}(x,y):=(x,x^ky),$$

where $x\in S^1 = \{z\in \Cc| |z|=1\}, y\in D^2=\{y\in\Cc||y|\le 1\}$$x\in S^1 = \{z\in \Cc| |z|=1\}, y\in D^2=\{y\in\Cc||y|\le 1\}$. Using this homeomorphism one can define a homeomorphism from $N_f$$N_f$ to $N$$N$:

$\displaystyle N = N\setminus L(S^1\times \mathring{D}^2)\cup_1 S^1\times D^2 \rightarrow N_{f_A}=N\setminus L(S^1 \times \mathring{D}^2)\cup_{f_A} S^1\times D^2$ $$\displaystyle N = N\setminus L(S^1\times \mathring{D}^2)\cup_1 S^1\times D^2 \rightarrow N_{f_A}=N\setminus L(S^1 \times \mathring{D}^2)\cup_{f_A} S^1\times D^2$$

given by the identity on the left component and $\bar{f_A}$$\bar{f_A}$ on the right component.

$\square$$\square$

Together with (link to comment about composition), this tells us, that $N_{f_A}$$N_{f_A}$ really only depends on the coset $A\cdot \left(\begin{array}{cc}1&*\\0&1\end{array}\right)$$A\cdot \left(\begin{array}{cc}1&*\\0&1\end{array}\right)$ (TODO check right or left coset). This coset is uniquely determined by the image $(p,q)$$(p,q)$ of $(1,0)$$(1,0)$ with $p$$p$ and $q$$q$ coprime.

The ratio $p/q$$p/q$ is called the surgery coefficient. (TODO what is the quotient good for ?)<++>

TODO does the result give different manifolds.

TODO does the result only depend on the isotopy class of the link.

Every compact (oriented /able, neccesary ?) 3-manifold might be obtained from $S^3$$S^3$ by a Dehn surgery along a link (TODO ref). Of course this does not satisfy to classify 3-manifolds without having a good classification of links in $S^3$$S^3$.

5 References

[Scott1983], [Thurston1997], [Hatcher2000], [Hempel1976]

• [Hatcher2000] A. Hatcher, Basic Topology of 3-Manifolds, Unpublished notes.

• [Hempel1976] J. Hempel, $3$$3$-Manifolds, Princeton University Press, Princeton, N. J., 1976. MR0415619 (54 #3702) Zbl 1058.57001
• [Scott1983] P. Scott, The geometries of $3$$3$-manifolds, Bull. London Math. Soc. 15 (1983), no.5, 401–487. MR705527 (84m:57009) Zbl 0662.57001
• [Thurston1997] W. P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton University Press, Princeton, NJ, 1997. MR1435975 (97m:57016) Zbl 0873.57001