Lesson 00

How to use this course

William Thurston changed topology twice. In the early 1970s he proved so many theorems about foliations that the subject seemed finished. Then, from the mid-1970s, he showed that most three-dimensional manifolds carry a hyperbolic geometry, and he conjectured that every one of them decomposes into geometric pieces. That conjecture, proved by Grigori Perelman in 2002–03, contains the Poincaré conjecture. Along the way he reshaped the study of surfaces, Kleinian groups, one-dimensional dynamics and complex dynamics, and wrote influentially about how mathematicians understand and communicate.

Prerequisites for Part II

Point-set and algebraic topology: fundamental group, covering spaces, homology.
Differential forms and the Frobenius theorem (Lesson 04).
The hyperbolic plane and Möbius transformations.
Helpful but not required: Riemannian geometry, Lie groups, one complex variable (Lesson 09).

Each technical lesson begins with an italic paragraph for the general reader, then gives definitions, precise statements, sketches of the key ideas, and exercises with folded solutions. Thurston himself cared most about the mental pictures behind proofs, so the sketches try to convey those pictures.

Reference

Timeline

1946	Born 30 October in Washington, D.C., to Margaret Thurston, a seamstress, and Paul Thurston, an aeronautical engineer.
1964–67	Member of the first class of New College, Sarasota, Florida. BA 1967.
1967	Graduate study at the University of California, Berkeley.
1972	PhD under Morris Hirsch: Foliations of three-manifolds which are circle bundles. Godbillon–Vey invariant takes all real values. Lectures at Orsay.
1972–73	Institute for Advanced Study, as assistant to John Milnor.
1973	Assistant professor at MIT.
1974	Full professor at Princeton. Invited address at the ICM in Vancouver. Sloan Fellowship.
1976	Every closed manifold of Euler characteristic zero has a codimension-one foliation. Veblen Prize, shared with James Simons. Circulates his work on surfaces and their diffeomorphisms.
1976–77	Orsay seminar on his surface theory, published in Astérisque in 1979.
1978	Princeton lecture notes The Geometry and Topology of Three-Manifolds begin to circulate. ICM speaker in Helsinki. Fellow of the American Academy of Arts and Sciences.
1979	Alan T. Waterman Award.
1982	“Three dimensional manifolds, Kleinian groups and hyperbolic geometry” in the Bulletin of the AMS, with 24 open problems. Fields Medal announced in Warsaw.
1983	Fields Medal presented at the delayed ICM in Warsaw. Elected to the National Academy of Sciences.
1985	Conjectures that circle packings converge to the Riemann map; proved by Rodin and Sullivan in 1987.
1986	Hyperbolic structures on 3-manifolds I in the Annals; A norm for the homology of 3-manifolds in the AMS Memoirs.
1988	Surface diffeomorphisms paper in the Bulletin; Milnor–Thurston kneading theory published.
1991	Professor at Berkeley.
1992–97	Director of the Mathematical Sciences Research Institute.
1994	“On proof and progress in mathematics” in the Bulletin of the AMS.
1996	Professor at the University of California, Davis.
1997	Three-Dimensional Geometry and Topology, Volume 1, edited by Silvio Levy.
2002–03	Perelman posts proofs of the geometrization conjecture.
2003	Jacob Gould Schurman Professor at Cornell, in mathematics and computer science.
2005	First AMS Book Prize, for Three-Dimensional Geometry and Topology.
2012	January: AMS Steele Prize for Seminal Contribution to Research. Dies 21 August in Rochester, New York, of melanoma, aged 65.

Part I · Formation

Lesson 01

Washington to Sarasota, 1946–1967

William Paul Thurston was born in Washington, D.C., on 30 October 1946. His mother, Margaret Thurston (née Martt), was a seamstress; his father, Paul Thurston, was an aeronautical engineer with a background in physics. Neither parent was a mathematician.

He was born with strabismus, a misalignment of the eyes that impaired his depth perception. When he was a toddler his mother worked with him on picture books, training him to reconstruct three-dimensional scenes from flat images. Thurston believed that this deliberate training in spatial reasoning helped shape his later geometric sense.

New College

In 1964 he entered New College in Sarasota, Florida, as a member of its first class. The college was experimental, with few requirements and much independent study. He graduated in 1967. His undergraduate thesis developed an intuitionist foundation for topology, a sign of an early interest in what it means to understand a mathematical object, which returned decades later in his essays on proof.

DiscussionThurston’s mathematics is famous for its visual intuition, yet he grew up with impaired depth perception. What does this suggest about the difference between seeing a picture and understanding a space?

Lesson 02

Berkeley and foliations, 1967–1972

Thurston began graduate study at the University of California, Berkeley, in 1967, at a time when the campus was a centre of political protest and of geometric topology. He worked with Morris Hirsch, and was also influenced by Stephen Smale. Hirsch directed him toward the theory of foliations: decompositions of a manifold into submanifolds, called leaves, that fit together locally like the pages of a book.

His 1972 dissertation, Foliations of three-manifolds which are circle bundles, studied codimension-one foliations of 3-manifolds that fibre over surfaces with circle fibres. He soon went far beyond it. A short paper of 1972 showed that the Godbillon–Vey invariant, a characteristic class of codimension-one foliations, can take every real value on the 3-sphere, so foliations come in uncountably many cobordism classes. That year he lectured at Orsay on foliations of higher codimension, and attended a foliations meeting at Plans-sur-Bex in Switzerland. John Milnor, who had received his preprints, wrote him a five-page letter about them in November 1971, and Thurston spent 1972–73 at the Institute for Advanced Study as Milnor’s assistant.

In the next four years Thurston proved the main existence theorems of the field (Lesson 04). Colleagues described the effect as overwhelming. Students stopped entering the subject because, it seemed, Thurston had already done everything. He later regretted this and drew lessons from it about how mathematical communities work.

Lesson 03

The turn to geometry, 1972–1978

After a year at the IAS as Milnor’s assistant, Thurston became an assistant professor at MIT in 1973. In 1974, at 27, he was appointed a full professor at Princeton. He gave an invited address at the International Congress in Vancouver that year, and in 1976 he shared the Veblen Prize in geometry with James Simons.

In the mid-1970s his interests moved from foliations to surfaces and 3-manifolds. Studying how a diffeomorphism of a surface acts on curves, he found the classification now called Nielsen–Thurston, together with a compactification of Teichmüller space by measured foliations. He circulated the results in 1976. The work was so compressed that a seminar at Orsay, organized by Valentin Poénaru with Albert Fathi, François Laudenbach and others, spent 1976–77 writing out the proofs. Thurston helped them at a session at Plans-sur-Bex in 1978, and the seminar appeared in Astérisque in 1979.

The decisive step came from an example. Thurston decomposed the complement of the figure-eight knot into two regular ideal tetrahedra and saw that it carries a complete hyperbolic metric. Testing many examples, he became convinced that hyperbolic structures are the rule rather than the exception. He taught a course at Princeton on the geometry and topology of 3-manifolds. Its notes, typed and mailed in instalments from 1978, became the most influential unpublished text in topology (Lesson 11).

Part II · The mathematics

Lesson 04

Foliations

For the general reader. A foliation slices a space into layers, the way an onion is made of shells or a book of pages. The question is which spaces can be sliced this way, and how many different slicings exist. Thurston answered both questions in the early 1970s, in a run of theorems that settled the main problems of the subject.

Goals

Define codimension-one foliations through integrable 1-forms.
Construct the Godbillon–Vey class and check that it is well defined.
State Thurston’s existence and stability theorems.

Definitions

A codimension-\(q\) foliation of an \(n\)-manifold \(M\) is a decomposition into connected, injectively immersed \((n-q)\)-dimensional leaves, locally modelled on the slices \(\mathbb R^{n-q}\times\{y\}\) of \(\mathbb R^n\). A transversely oriented codimension-one foliation is given, up to multiplication by a positive function, by a non-vanishing 1-form \(\omega\) with \(\ker\omega=T\mathcal F\). By Frobenius, the plane field \(\ker\omega\) is tangent to a foliation if and only if

\[ \omega\wedge d\omega=0, \qquad\text{equivalently}\qquad d\omega=\omega\wedge\eta \ \text{ for some 1-form } \eta. \]

The basic example is the Reeb foliation of \(S^3\): two solid tori, each foliated by one torus leaf on its boundary and planes spiralling toward it inside.

The Godbillon–Vey invariant

For \(M\) of dimension 3, the 3-form \(\eta\wedge d\eta\) is closed and its cohomology class \(GV(\mathcal F)\in H^3(M;\mathbb R)\) depends only on \(\mathcal F\) (Exercise 4.2). For closed oriented \(M^3\), it is the number \(\int_M\eta\wedge d\eta\).

Theorem (Thurston, 1972)

For every real number \(r\) there is a smooth codimension-one foliation \(\mathcal F_r\) of \(S^3\) with \(GV(\mathcal F_r)=r\). Hence the cobordism group of codimension-one foliations of 3-manifolds maps onto \(\mathbb R\).

Roussarie had found a non-zero invariant for the horocyclic foliation of a compact quotient of \(\mathrm{SL}_2(\mathbb R)\). Thurston built families of foliations from hyperbolic geometry on which the invariant varies continuously, then transplanted them into \(S^3\). Thurston also gave a geometric reading of the invariant as a measure of helical wobble, how leaves twist as they spread apart.

Existence and stability

Theorem (Thurston, 1974 and 1976)

(1) Let \(q\ge2\). If the normal bundle of a codimension-\(q\) plane field on \(M\) carries a Haefliger \(\Gamma_q\)-structure, then the plane field is homotopic to the tangent field of a foliation. The existence of foliations of codimension at least two is therefore a problem in homotopy theory.

(2) A closed manifold admits a smooth codimension-one foliation if and only if its Euler characteristic is zero.

The necessity of \(\chi(M)=0\) in (2) is classical: a foliation gives a line field transverse to the leaves, and a nowhere-zero vector field exists only when \(\chi=0\) (after passing to a double cover if needed). The difficulty was sufficiency in all dimensions, which Thurston proved by first making a plane field integrable away from a finite collection of spheres and then filling the holes with foliated pieces. He also generalized Reeb’s stability theorem: if a transversely oriented \(C^1\) codimension-one foliation of a closed connected manifold has a compact leaf \(L\) with \(H^1(L;\mathbb R)=0\), then every leaf is diffeomorphic to \(L\) and the manifold fibres over the circle with fibre \(L\).

Exercise 4.1Which closed surfaces admit a foliation by curves? Give one on each.

Show solution

By (2), exactly those with \(\chi=0\): the torus and the Klein bottle. On the torus \(\mathbb R^2/\mathbb Z^2\), take lines of any fixed slope. On the Klein bottle, the quotient of \(\mathbb R^2\) by \((x,y)\mapsto(x+1,-y)\) and \((x,y)\mapsto(x,y+1)\), the horizontal lines \(y=\text{const}\) are preserved and descend to a foliation.

Exercise 4.2Suppose \(d\omega=\omega\wedge\eta\). (a) Show that \(\omega\wedge d\eta=0\). (b) Show that replacing \(\eta\) by \(\eta+f\omega\) changes \(\eta\wedge d\eta\) by the exact form \(d(f\,d\omega)\).

Show solution

(a) \(0=d(d\omega)=d\omega\wedge\eta-\omega\wedge d\eta=\omega\wedge\eta\wedge\eta-\omega\wedge d\eta=-\omega\wedge d\eta\).

(b) With \(\eta'=\eta+f\omega\), \(d\eta'=d\eta+df\wedge\omega+f\,\omega\wedge\eta\). Expanding \(\eta'\wedge d\eta'\), the terms containing \(\omega\) twice or \(\eta\) twice vanish, leaving \(\eta\wedge d\eta+\eta\wedge df\wedge\omega+f\,\omega\wedge d\eta\). The last term is 0 by (a). The middle term equals \(df\wedge\omega\wedge\eta=df\wedge d\omega=d(f\,d\omega)\). Rescaling \(\omega\) by a positive function \(e^u\) changes \(\eta\) by \(-du\) modulo \(\omega\), and \(du\wedge d\eta=d(u\,d\eta)\) is exact too.

Lesson 05

Surfaces and their diffeomorphisms

For the general reader. Stretch a rubber surface and glue it back onto itself. Up to continuous deformation, Thurston showed, only three things can happen: the map eventually repeats, it preserves some family of curves so the surface can be cut into simpler pieces, or it stretches the surface uniformly in one direction while compressing it in another. The third kind, pseudo-Anosov maps, turned out to be the typical case and the source of hyperbolic geometry in three dimensions.

Goals

Define measured foliations and pseudo-Anosov maps.
State the Nielsen–Thurston classification and the Thurston compactification.
Compute with the Anosov map of the torus.

Measured foliations

Let \(S\) be a closed orientable surface of genus \(g\ge2\). A measured foliation \((\mathcal F,\mu)\) is a foliation with finitely many singularities of \(k\)-prong type, \(k\ge3\), together with a transverse measure \(\mu\) invariant under sliding along leaves. Each simple closed curve \(\gamma\) has an intersection number \(i(\mu,\gamma)=\inf\int_{\gamma'}d\mu\) over curves \(\gamma'\) isotopic to \(\gamma\). Up to isotopy and Whitehead moves, measured foliations form a space \(\mathcal{MF}(S)\cong\mathbb R^{6g-6}\setminus\{0\}\), and its projectivization \(\mathcal{PMF}(S)\) is a sphere of dimension \(6g-7\). Simple closed curves, with weights, are dense.

Theorem (Thurston)

Teichmüller space \(\mathcal T(S)\cong\mathbb R^{6g-6}\), together with \(\mathcal{PMF}(S)\) as boundary, is a closed ball of dimension \(6g-6\), and the action of the mapping class group extends continuously to it.

The classification

A homeomorphism \(f:S\to S\) is pseudo-Anosov if there are transverse measured foliations \((\mathcal F^s,\mu^s)\) and \((\mathcal F^u,\mu^u)\) and a number \(\lambda>1\), the stretch factor, with

\[ f(\mathcal F^u,\mu^u)=(\mathcal F^u,\lambda\mu^u), \qquad f(\mathcal F^s,\mu^s)=(\mathcal F^s,\lambda^{-1}\mu^s). \]

Theorem (Nielsen–Thurston classification)

Every homeomorphism of \(S\) is isotopic to a homeomorphism \(f\) that is (i) periodic, \(f^n=\mathrm{id}\) for some \(n\ge1\); or (ii) reducible, preserving a non-empty union of disjoint essential simple closed curves; or (iii) pseudo-Anosov. Pseudo-Anosov maps are neither periodic nor reducible.

Idea of the proof. Apply Brouwer’s fixed point theorem to the action of \(f\) on the closed ball \(\mathcal T(S)\cup\mathcal{PMF}(S)\). An interior fixed point is a hyperbolic metric preserved by \(f\) up to isotopy, so \(f\) is isotopic to an isometry, which is periodic. A fixed point on the boundary is a projective measured foliation \([\mu]\) with \(f(\mu)=\lambda\mu\). If \(\mu\) has a closed leaf, \(f\) is reducible. Otherwise one shows \(\lambda\ne1\), and \(\mu\) together with a second fixed foliation gives the pseudo-Anosov structure. For pseudo-Anosov \(f\) and every curve \(\gamma\), the lengths grow like \(\lambda^n\), and \(\lambda\) is an algebraic integer.

Thurston also proved that the mapping torus of a pseudo-Anosov map is hyperbolic (Lesson 06), linking the dynamics of surfaces to the geometry of 3-manifolds.

Exercise 5.1The matrix \(A=\begin{pmatrix}2&1\\1&1\end{pmatrix}\) acts on the torus \(\mathbb R^2/\mathbb Z^2\). Find its stretch factor and the slopes of its invariant foliations, and explain why every leaf is dense.

Show solution

The characteristic polynomial is \(t^2-3t+1\), with roots \(\lambda^{\pm1}\), \(\lambda=\frac{3+\sqrt5}{2}\). The eigenvectors \((1,\frac{\sqrt5-1}{2})\) and \((1,-\frac{1+\sqrt5}{2})\) have irrational slopes, so the lines in these directions project to dense leaves on the torus. The foliations by lines in these directions, with transverse Lebesgue measure, are expanded and contracted by \(\lambda\). This is the Anosov model that pseudo-Anosov maps generalize, allowing singularities.

Exercise 5.2On the torus, let \(a\) and \(b\) be the curves of slope \(0\) and \(\infty\), and let \(T_a\) be the Dehn twist with matrix \(\begin{pmatrix}1&1\\0&1\end{pmatrix}\). Compute the geometric intersection number \(i(T_a^n(b),b)\), and compare its growth with that of a pseudo-Anosov map.

Show solution

\(T_a^n\) sends the vector \((0,1)\) of \(b\) to \((n,1)\). For curves of slopes \((p,q)\) and \((r,s)\) on the torus, \(i=|ps-qr|\), so \(i(T_a^n b,b)=|n\cdot1-1\cdot0|=n\). Growth is linear, as for any reducible twist. For \(A\) in Exercise 5.1, \(i(A^n b,b)\) grows like \(\lambda^n\), exponentially.

Lesson 06

Hyperbolic 3-manifolds

For the general reader. In hyperbolic space the angles of a triangle add up to less than 180 degrees, and space expands exponentially. Before Thurston, three-dimensional spaces carrying such a geometry looked rare and special. Thurston showed that they are everywhere: most knot complements are hyperbolic, and almost every way of filling in a hyperbolic knot complement produces another hyperbolic space.

Goals

Build the figure-eight knot complement from ideal tetrahedra.
Compute volumes with the Lobachevsky function.
State the hyperbolic Dehn surgery theorem and the structure of the volume spectrum.

Ideal tetrahedra

In the upper half-space model \(\mathbb H^3\), an ideal tetrahedron has its four vertices on the sphere at infinity \(\hat{\mathbb C}\). Up to isometry its vertices can be placed at \(0,1,\infty,z\) with \(\operatorname{Im}z>0\), and the complex number \(z\) is its shape. Opposite edges have equal dihedral angles, and the three angles \(\alpha,\beta,\gamma\) at each vertex sum to \(\pi\). The volume is

\[ \operatorname{Vol}=\Lambda(\alpha)+\Lambda(\beta)+\Lambda(\gamma), \qquad \Lambda(\theta)=-\int_0^\theta\log|2\sin t|\,dt. \]

The regular ideal tetrahedron has all angles \(\pi/3\) and volume \(v_3=3\Lambda(\pi/3)\approx1.0149416\), the largest of any hyperbolic tetrahedron.

Theorem (Thurston, 1977)

The complement of the figure-eight knot in \(S^3\) is obtained by gluing the faces of two regular ideal tetrahedra in pairs. It carries a complete hyperbolic metric of finite volume \(2v_3\approx2.0298832\).

Gluing equations demand that the dihedral angles around each edge sum to \(2\pi\), and that the shapes multiply to 1 around each edge. In the figure-eight decomposition each of the two edge classes receives six dihedral angles of \(\pi/3\). Robert Riley had found the hyperbolic structure algebraically a little earlier, through a representation of the knot group into \(\mathrm{PSL}_2(\mathbb Z[\omega])\); Thurston’s geometric picture showed how to generalize.

Dehn surgery and volume

Let \(M\) be a hyperbolic 3-manifold of finite volume with one cusp, a neighbourhood of an end homeomorphic to \(T^2\times[0,\infty)\). Cutting off the cusp and gluing in a solid torus so that a meridian disc bounds the curve of slope \(p/q\) gives a closed manifold \(M(p/q)\), the Dehn filling.

Theorem (Thurston’s hyperbolic Dehn surgery theorem)

All but finitely many Dehn fillings \(M(p/q)\) are hyperbolic. Their volumes are strictly less than \(\operatorname{Vol}(M)\) and converge to it as \(p^2+q^2\to\infty\).

The proof deforms the complete structure through incomplete hyperbolic structures, parametrized near the complete one by a neighbourhood of a point in \(\mathbb C\); generic deformations complete to cone manifolds, and when the cone angle reaches \(2\pi\) one obtains a closed manifold. For the figure-eight knot exactly ten fillings fail to be hyperbolic: \(\infty\) (which gives \(S^3\)), \(0\), \(\pm1\), \(\pm2\), \(\pm3\) and \(\pm4\).

Theorem (Jørgensen–Thurston)

The set of volumes of complete hyperbolic 3-manifolds is a well-ordered subset of \(\mathbb R\) of order type \(\omega^\omega\). Only finitely many manifolds share any given volume.

Mostow–Prasad rigidity says that a finite-volume hyperbolic structure, when it exists, is unique, so volume is a topological invariant. The smallest closed hyperbolic manifold is the Weeks manifold, of volume about 0.9427, as Gabai, Meyerhoff and Milley proved in 2009.

Exercise 6.1The edges of the tetrahedron with vertices \(0,1,\infty,z\) carry shape parameters \(z\), \(z'=\frac{1}{1-z}\) and \(z''=\frac{z-1}{z}\). Show that \(zz'z''=-1\), and that for the regular tetrahedron \(z=z'=z''=e^{i\pi/3}\).

Show solution

\(zz'z''=z\cdot\frac{1}{1-z}\cdot\frac{z-1}{z}=\frac{z-1}{1-z}=-1\). For \(z=e^{i\pi/3}=\frac12+\frac{\sqrt3}{2}i\), \(1-z=\frac12-\frac{\sqrt3}{2}i=e^{-i\pi/3}\), so \(z'=e^{i\pi/3}\), and \(z''=1-\frac1z=1-e^{-i\pi/3}=e^{i\pi/3}\). The arguments of \(z,z',z''\) are the three dihedral angles, here all \(\pi/3\), summing to \(\pi\) because the product is \(-1\).

Exercise 6.2Count the edges of the figure-eight decomposition from Euler characteristic, given that the two tetrahedra have 12 edges in total and that the cusp cross-section is a torus triangulated by the 8 vertex links.

Show solution

Each ideal vertex contributes a triangle to the cusp torus, giving 8 triangles, hence 12 edges of the torus triangulation and, by \(\chi(T^2)=0\), \(V-12+8=0\), so 4 vertices. Each vertex of the cusp triangulation is an end of an edge of the decomposition, and each edge has two ends, so there are 2 edges. Twelve tetrahedron edges fall into 2 classes of 6, which is why each class receives six angles of \(\pi/3\), summing to \(2\pi\).

Lesson 07

The geometrization conjecture

For the general reader. Every closed surface has a geometry: round, flat or hyperbolic. Thurston proposed that every three-dimensional space, once cut along certain spheres and tori, breaks into pieces each carrying one of exactly eight geometries. He proved this for a huge class of spaces. Perelman’s proof of the full conjecture in 2003 also solved the Poincaré conjecture.

Goals

List the eight model geometries.
State the geometrization conjecture and Thurston’s hyperbolization theorems.
See why geometrization implies the Poincaré conjecture.

The eight geometries

A model geometry is a simply connected manifold \(X\) with a transitive Lie group \(G\) of isometries with compact stabilizers, maximal among such groups, and admitting a quotient of finite volume. Thurston proved that in dimension three there are exactly eight:

\[ S^3,\quad \mathbb E^3,\quad \mathbb H^3,\quad S^2\times\mathbb R,\quad \mathbb H^2\times\mathbb R,\quad \widetilde{\mathrm{SL}_2(\mathbb R)},\quad \mathrm{Nil},\quad \mathrm{Sol}. \]

Six of these describe Seifert fibred spaces, Sol describes certain torus bundles over the circle, and hyperbolic geometry covers everything else.

Geometrization conjecture (Thurston, 1982; proved by Perelman, 2002–03)

Let \(M\) be a closed orientable 3-manifold. Cut \(M\) along essential spheres (the prime decomposition of Kneser and Milnor) and then along a minimal family of incompressible tori (the JSJ decomposition of Jaco, Shalen and Johannson). Then the interior of each resulting piece admits a complete, finite-volume geometric structure modelled on one of the eight geometries.

Theorem (Thurston’s hyperbolization theorem for Haken manifolds)

Let \(M\) be a compact, orientable, irreducible, atoroidal 3-manifold that contains an incompressible surface (is Haken), whose boundary is empty or a union of tori, and which is not Seifert fibred. Then the interior of \(M\) admits a complete hyperbolic metric of finite volume.

The proof cuts \(M\) along incompressible surfaces down to balls and rebuilds it step by step. At each gluing one must find a fixed point of the skinning map on Teichmüller space; for manifolds that fibre over the circle a separate argument, the double limit theorem, is needed. Thurston called this result the Monster Theorem. Complete written proofs, by Morgan, McMullen, Otal and Kapovich, took nearly twenty years. Thurston also announced in 1981 the orbifold theorem, geometrizing 3-orbifolds with one-dimensional singular locus; written proofs appeared around 2000.

The 1982 problem list

Thurston’s 1982 Bulletin article posed 24 questions on 3-manifolds and Kleinian groups. They organized the field for thirty years. The last major ones, the virtual Haken and virtual fibring conjectures, were settled by Ian Agol in 2012, building on Daniel Wise, a few months before Thurston’s death.

Exercise 7.1Match each manifold with its geometry: the 3-torus; \(S^2\times S^1\); a lens space \(L(p,q)\); \(\Sigma_g\times S^1\) with \(g\ge2\); the unit tangent bundle of a closed hyperbolic surface; the Heisenberg nilmanifold; the mapping torus of \(\begin{pmatrix}2&1\\1&1\end{pmatrix}\) on \(T^2\); the figure-eight knot complement.

Show solution

\(T^3\): \(\mathbb E^3\). \(S^2\times S^1\): \(S^2\times\mathbb R\). \(L(p,q)\): \(S^3\). \(\Sigma_g\times S^1\): \(\mathbb H^2\times\mathbb R\). Unit tangent bundle: \(\widetilde{\mathrm{SL}_2(\mathbb R)}\), since it is \(\mathrm{PSL}_2(\mathbb R)/\Gamma\). Heisenberg nilmanifold: Nil. Anosov mapping torus: Sol. Figure-eight complement: \(\mathbb H^3\).

Exercise 7.2Explain why geometrization implies the Poincaré conjecture: a closed simply connected 3-manifold is homeomorphic to \(S^3\).

Show solution

Write \(M=M_1\#\cdots\#M_k\) as a sum of primes. Since \(\pi_1(M)\) is the free product of the \(\pi_1(M_i)\), every summand is simply connected. A summand contains no incompressible torus, since that would inject \(\mathbb Z^2\) into the trivial group, so it is not cut further and carries one of the eight geometries: \(M_i\cong X/\Gamma\) with \(\Gamma\cong\pi_1(M_i)=1\). Then \(X=M_i\) is compact, and among the eight model spaces only \(S^3\) is compact. Each summand is \(S^3\), and so is their connected sum.

Lesson 08

The Thurston norm and fibrations

For the general reader. A 3-manifold contains many surfaces. Thurston measured the simplest surface in each homology class and found that this measure is a norm whose unit ball is a polyhedron. The faces of the polyhedron record all the ways the manifold can be swept out by a moving surface, so that one fibration usually comes with infinitely many others.

Goals

Define the Thurston norm and prove it is a seminorm on integral classes.
State the polyhedrality and fibred-face theorems.
Compute examples.

Definition

Let \(M\) be a compact orientable 3-manifold. For a compact surface \(S\) with components \(S_i\), set \(\chi_-(S)=\sum_i\max(0,-\chi(S_i))\). For \(\phi\in H^1(M;\mathbb Z)\cong H_2(M,\partial M;\mathbb Z)\), define

\[ x(\phi)=\min\{\chi_-(S): S\subset M \text{ properly embedded, oriented, } [S]\text{ dual to }\phi\}. \]

Theorem (Thurston, 1976; published 1986)

(1) \(x\) is a seminorm on \(H^1(M;\mathbb Z)\): \(x(n\phi)=|n|x(\phi)\) and \(x(\phi+\psi)\le x(\phi)+x(\psi)\). It extends to a continuous seminorm on \(H^1(M;\mathbb R)\), and is a norm if \(M\) contains no essential surfaces of non-negative Euler characteristic.

(2) The unit ball \(B_x\) is a polyhedron: the dual ball is the convex hull of finitely many integral points \(\pm\alpha_i\in H_1(M;\mathbb Z)\), and \(B_x=\{\phi:|\phi(\alpha_i)|\le1\ \forall i\}\).

(3) If \(M\) fibres over the circle with fibre \(S\), \(\chi(S)<0\), then \([S]\) lies in the open cone over a top-dimensional face \(F\) of \(B_x\), and every integral class in that open cone is represented by a fibration. Such faces are called fibred faces.

Subadditivity comes from cut-and-paste: surfaces representing \(\phi\) and \(\psi\), in general position, can be resolved along their intersection curves into an embedded surface representing \(\phi+\psi\) with \(\chi_-\) at most the sum. The norm is Thurston’s; its relation to the Alexander polynomial, and the Teichmüller polynomial that computes the stretch factors of all monodromies in a fibred face, were found later by McMullen.

Exercise 8.1Let \(M=\Sigma_g\times S^1\) with \(g\ge2\). For the class \(\phi\) dual to \(\Sigma_g\times\{pt\}\) and the class \(\psi\) dual to \(c\times S^1\), \(c\) a non-separating curve, compute \(x(\phi)\) and \(x(\psi)\). Is \(x\) a norm?

Show solution

The fibre \(\Sigma_g\) is norm-minimizing (fibres always are), so \(x(\phi)=2g-2\). The torus \(c\times S^1\) has \(\chi=0\), so \(x(\psi)=0\). Since \(\psi\ne0\), \(x\) is only a seminorm; the essential tori are the obstruction.

Exercise 8.2The figure-eight knot complement fibres over the circle with fibre a once-punctured torus and monodromy \(\begin{pmatrix}2&1\\1&1\end{pmatrix}\). Find \(x\) on \(H^1\cong\mathbb Z\), describe \(B_x\), and give the stretch factor of the monodromy.

Show solution

The fibre has \(\chi=2-2-1=-1\), so \(x(\pm\phi)=1\) for a generator and \(x(n\phi)=|n|\). The unit ball is the interval \([-1,1]\); its two endpoints are fibred faces. The stretch factor is \(\frac{3+\sqrt5}{2}\), the larger root of \(t^2-3t+1\).

Lesson 09

One-dimensional and complex dynamics

For the general reader. Iterating a simple function, such as \(x\mapsto ax(1-x)\), can produce order or chaos depending on \(a\). With John Milnor, Thurston gave a complete combinatorial description of these systems and proved that chaos increases steadily with the parameter. He also found a criterion deciding when a topological picture of a branched map of the sphere comes from a genuine rational function, a theorem that became a basic tool of complex dynamics.

Goals

Understand kneading sequences and the monotonicity of entropy.
State Thurston’s characterization of rational maps.
Compute postcritical sets.

Kneading theory

A unimodal map \(f:[0,1]\to[0,1]\) increases up to a critical point \(c\) and then decreases. Its kneading sequence records, for each \(n\ge1\), whether \(f^n(c)\) lies left of, at, or right of \(c\). Milnor and Thurston showed, in work circulated from 1977 and published in 1988, that the kneading sequence determines the combinatorics of all orbits, and they computed the topological entropy \(h(f)\) from it.

Theorem (Milnor–Thurston)

For the quadratic family \(q_a(x)=ax(1-x)\), \(0\le a\le4\), the topological entropy \(h(q_a)\) is a continuous, non-decreasing function of \(a\), rising from 0 to \(\log2\).

The proof of monotonicity uses complex analysis: a real quadratic map with a given periodic kneading behaviour is unique, which follows from Thurston’s rigidity for postcritically finite maps.

Thurston’s characterization of rational maps

Let \(f:S^2\to S^2\) be an orientation-preserving branched covering of degree \(d\ge2\) whose postcritical set \(P_f=\bigcup_{n\ge1}f^n(\text{critical points})\) is finite. A multicurve \(\Gamma\) is a finite set of disjoint, non-homotopic, essential simple closed curves in \(S^2\setminus P_f\), essential meaning that each side contains at least two points of \(P_f\). It is invariant if every essential component of \(f^{-1}(\gamma)\), \(\gamma\in\Gamma\), is homotopic in \(S^2\setminus P_f\) to a curve of \(\Gamma\). Its Thurston matrix is

\[ M_{\gamma\delta}=\sum_{\alpha}\frac{1}{\deg\big(f:\alpha\to\delta\big)}, \]

summing over the components \(\alpha\) of \(f^{-1}(\delta)\) homotopic to \(\gamma\).

Theorem (Thurston; proof by Douady and Hubbard, 1993)

Suppose the orbifold of \(f\) is hyperbolic. Then \(f\) is equivalent (by homotopy relative to \(P_f\)) to a rational map if and only if every invariant multicurve has leading eigenvalue \(\lambda(\Gamma)<1\). When it exists, the rational map is unique up to Möbius conjugacy.

The proof iterates a map on Teichmüller space of the punctured sphere \(S^2\setminus P_f\), pulling complex structures back by \(f\). Either the iteration converges to a fixed point, giving the rational map, or some curves become short and form an obstruction. The same strategy, iteration on Teichmüller space, appears in Thurston’s hyperbolization proof.

Exercise 9.1Find the postcritical sets of \(z^2-1\) and \(z^2+i\) (include \(\infty\)).

Show solution

The critical points are \(0\) and \(\infty\). For \(z^2-1\): \(0\mapsto-1\mapsto0\), so \(P_f=\{0,-1,\infty\}\). For \(z^2+i\): \(0\mapsto i\mapsto-1+i\mapsto-i\mapsto-1+i\), so \(P_f=\{i,-1+i,-i,\infty\}\), and the critical point is strictly preperiodic.

Exercise 9.2Show that \(q_4(x)=4x(1-x)\) is conjugate to the tent map \(T(y)=1-|2y-1|\) by \(x=\sin^2(\pi y/2)\), and deduce \(h(q_4)=\log2\).

Show solution

With \(x=\sin^2\theta\), \(\theta=\pi y/2\), we get \(4x(1-x)=4\sin^2\theta\cos^2\theta=\sin^2 2\theta=\sin^2(\pi y)\), and \(\sin^2(\pi y)=\sin^2\big(\tfrac{\pi}{2}T(y)\big)\) because \(\sin^2\) is symmetric about \(\pi/2\). The tent map has \(2^n\) laps for \(T^n\), and by Misiurewicz–Szlenk the entropy is \(\lim\frac1n\log(\text{laps})=\log2\). Entropy is a conjugacy invariant.

Lesson 10

Circle packings and triangulations

For the general reader. Arrange coins on a table so that some touch. Thurston saw that such packings behave like holomorphic functions, and that finer and finer packings can approximate the conformal maps of complex analysis. He also used hyperbolic geometry to count the ways of building a convex polyhedron from equilateral triangles.

Goals

State the Koebe–Andreev–Thurston theorem and the Rodin–Sullivan convergence theorem.
Understand Thurston’s shape space of polyhedra.
Use Euler’s formula on triangulations with small vertex degree.

Circle packings

Theorem (Koebe 1936; Andreev 1970; Thurston)

For every triangulation \(K\) of the 2-sphere there is a packing of circles on \(S^2\) with disjoint interiors whose tangency graph is the 1-skeleton of \(K\). It is unique up to Möbius transformations.

Thurston rediscovered the theorem in his Princeton notes through Andreev’s work on hyperbolic polyhedra, and proposed in 1985 that it gives a discrete Riemann mapping theorem. Fill a simply connected domain \(\Omega\) with a hexagonal packing of circles of radius \(\varepsilon\). The theorem gives a combinatorially equal packing filling the unit disc, and hence a map from \(\Omega\) to the disc sending centres to centres.

Theorem (Rodin–Sullivan, 1987; Thurston’s conjecture)

As \(\varepsilon\to0\), suitably normalized, these maps converge uniformly on compact subsets of \(\Omega\) to the Riemann map \(\Omega\to\mathbb D\).

Shapes of polyhedra

A convex polyhedron with \(n\) vertices, whose faces are all equilateral triangles, has cone angle \(\frac{\pi}{3}d_v\le2\pi\) at a vertex of degree \(d_v\), so every degree is at most 6. Thurston studied the space of flat metrics on the sphere with \(n\) cone points of prescribed angles less than \(2\pi\).

Theorem (Thurston; published 1998)

The space of such metrics of fixed area, with fixed cone angles, carries a natural Kähler metric locally isometric to complex hyperbolic space \(\mathbb{CH}^{n-3}\), and its metric completion is a complex hyperbolic cone manifold of finite volume. Triangulations of the sphere with all vertex degrees at most 6 correspond to lattice points in these spaces.

Counting lattice points of a finite-volume complex hyperbolic orbifold gives the number of triangulations. For triangulations with twelve vertices of degree 5 and the rest of degree 6, the space has complex dimension \(12-3=9\), and the number of such triangulations with at most \(N\) triangles grows like a constant times \(N^{10}\). In a related paper with Daniel Sleator and Robert Tarjan (1988), he used hyperbolic volume to prove that the rotation distance between binary trees with \(n\) internal nodes reaches \(2n-6\) for large \(n\).

Exercise 10.1Let \(K\) be a triangulation of \(S^2\) with \(V\) vertices of degrees \(d_v\). Prove \(\sum_v(6-d_v)=12\). Deduce that if all degrees are 5 or 6, exactly twelve vertices have degree 5.

Show solution

Each face has three edges and each edge two faces, so \(2E=3F\). Each edge has two endpoints, so \(\sum d_v=2E\). Euler gives \(V-E+F=2\), hence \(V-E+\tfrac23E=2\), so \(E=3V-6\) and \(\sum d_v=6V-12\). Therefore \(\sum(6-d_v)=12\). With degrees in \(\{5,6\}\), each degree-5 vertex contributes 1, so there are twelve. The icosahedron (\(V=12\)) and the fullerene-type triangulations are examples.

Exercise 10.2Troyanov’s theorem says that for prescribed cone angles \(\theta_i\in(0,2\pi)\) with \(\sum(2\pi-\theta_i)=4\pi\), each choice of \(n\) distinct points on the Riemann sphere carries a unique flat cone metric of area 1 with those angles. Use this to recover the dimension \(n-3\), and check the curvature condition for a convex polyhedron with equilateral faces.

Show solution

By Troyanov, metrics of area 1 correspond to configurations of \(n\) distinct points on \(\hat{\mathbb C}\) modulo Möbius transformations, which is the moduli space \(\mathcal M_{0,n}\). Möbius maps are 3-transitive, so we may fix three points at \(0,1,\infty\), leaving \(n-3\) free complex coordinates. For the polyhedron, \(2\pi-\theta_v=\frac{\pi}{3}(6-d_v)\), and Exercise 10.1 gives \(\sum\frac{\pi}{3}(6-d_v)=\frac{\pi}{3}\cdot12=4\pi\), as Gauss–Bonnet requires.

Part III · Life and legacy

Lesson 11

Princeton, the Notes, the Fields Medal, 1974–1991

Thurston spent seventeen years at Princeton. His lecture notes, The Geometry and Topology of Three-Manifolds, were typed from his Princeton course and distributed in instalments from 1978 to anyone who asked. They contained the figure-eight example, hyperbolic Dehn surgery, the theory of orbifolds, Andreev’s theorem, the convergence of Kleinian groups, and pictures that topologists still redraw. A generation of mathematicians learned geometry from photocopies of them. In 1982 he published his program and 24 problems in the Bulletin of the AMS.

The Fields Medal was announced at the IMU General Assembly in Warsaw in August 1982 and presented in 1983, at the International Congress that had been postponed because of martial law in Poland. Alain Connes and Shing-Tung Yau were the other medallists. The citation credited him with revolutionizing the study of topology in two and three dimensions, and with proposing that a very large class of closed 3-manifolds carry a hyperbolic structure. He was elected to the National Academy of Sciences in 1983, after the American Academy of Arts and Sciences (1978) and the Alan T. Waterman Award (1979).

Students

He supervised more than thirty doctoral students, most at Princeton. Among them are Steven Kerckhoff (1978), who solved the Nielsen realization problem with Thurston’s earthquakes; William Floyd (1978); David Gabai (1980); William Goldman (1980); Lee Mosher (1983); Jeffrey Weeks (1985), who wrote the program SnapPea; Craig Hodgson (1986); Igor Rivin (1986); Richard Canary (1989); Yair Minsky (1989), who with Jeffrey Brock and Canary proved the ending lamination conjecture; Richard Kenyon (1990); Oded Schramm (1990); Richard Schwartz (1991); Benson Farb (1994); and Danny Calegari (2000). His son Dylan Thurston is also a mathematician.

DiscussionMuch of Thurston’s most influential work circulated as unpublished notes for years, and some proofs were written by others decades later. Weigh the benefits and costs of this way of working for the field and for the people who completed the proofs.

Lesson 12

MSRI, Davis, Cornell and legacy, 1991–2012

In 1991 Thurston returned to Berkeley as professor, and from 1992 to 1997 he directed the Mathematical Sciences Research Institute. He worked with the Geometry Center at the University of Minnesota on visual tools for mathematics; its films include Not Knot (1991), on hyperbolic knot complements, and Outside In (1994), which shows his method of turning a sphere inside out by corrugations. In 1996 he moved to the University of California, Davis, and in 2003 to Cornell as Jacob Gould Schurman Professor in mathematics and in computing and information science.

On proof and progress

In 1993 Arthur Jaffe and Frank Quinn criticized the publication of incomplete, speculative mathematics, citing the gap between Thurston’s announcements and written proofs. Thurston replied in 1994 with “On proof and progress in mathematics”. He argued that the goal of mathematics is human understanding, not a list of verified theorems, and described how his foliation work had left a field without people who understood it, while his 3-manifold notes had tried instead to build a community. The essay is still widely read. His book Three-Dimensional Geometry and Topology (1997), edited by Silvio Levy, won the first AMS Book Prize in 2005.

Final years

Perelman’s proof of geometrization, posted in 2002 and 2003 and verified over the following years, completed Thurston’s program. In January 2012 Thurston received the AMS Steele Prize for Seminal Contribution to Research, for the series of papers beginning with Hyperbolic structures on 3-manifolds I. He had been diagnosed with melanoma in 2011, and he died in Rochester, New York, on 21 August 2012, aged 65.

Legacy

3-manifolds: geometrization, hyperbolic Dehn surgery and the problem list framed the subject until Agol’s 2012 theorems resolved the last of the major questions.
Surfaces and Kleinian groups: measured laminations, earthquakes, train tracks and the Thurston compactification are the basic language of Teichmüller theory, used throughout Mirzakhani’s and McMullen’s work.
Geometric group theory: with Mikhail Gromov he is regarded as a founder; automatic groups (Epstein, Cannon, Holt, Levy, Paterson and Thurston, 1992) came from his questions.
Dynamics: kneading theory and the characterization of rational maps underlie modern real and complex dynamics.

Final projectChoose one theorem from Part II. Write a two-page account for a strong undergraduate that states it, works one example in full, and describes the mental picture that, in Thurston’s view, carries the proof.

Reference

Connections

People linked to William P. Thurston across this collection. A name without a link has no page here yet. See William P. Thurston in the network of all 337 lives →

Teachers and advisors

Morris Hirsch 1972

Students

Benson Farb 1994
Craig Hodgson 1986
Danny Calegari 2000
David Gabai 1980
Igor Rivin 1986
Jeffrey Weeks 1985
Lee Mosher 1983
Oded Schramm 1990
Richard Canary 1989
Richard Kenyon 1990
Richard Schwartz 1991
Silvio Levy 1985
Steven Kerckhoff 1978
William Floyd 1978
William Goldman 1980
Yair Minsky 1989

Influenced by

Influenced

Colleagues and collaborators

Dennis Sullivan
John Milnor
Mikhail Gromov
Adrien Douady
Daniel Sleator 1988
John H. Hubbard
Robert Tarjan 1988
Troels Jørgensen
Valentin Poénaru

Family

Dylan Thurston

Also mentioned in this life

Fields

Reference

Sources and further reading

Biography

J. J. O’Connor and E. F. Robertson, William Paul Thurston, MacTutor History of Mathematics.
William Thurston, Wikipedia.
Cornell Chronicle, World-renowned mathematician William Thurston dies at 65, August 2012.
Cornell Department of Mathematics, In memory of William P. Thurston.
Cornell Chronicle, Bill Thurston receives prestigious mathematics prize, January 2012.
J. Malkevitch, Remembering Bill Thurston (1946–2012), AMS Feature Column, 2017.
William P. Thurston, 1946–2012, memorial article, Notices AMS 62 (2015).
F. Laudenbach and A. Papadopoulos, W. P. Thurston and French mathematics, arXiv:1912.03115.
William Paul Thurston, Mathematics Genealogy Project.

Mathematics

W. P. Thurston, Noncobordant foliations of \(S^3\), Bull. AMS 78 (1972).
W. P. Thurston, A generalization of the Reeb stability theorem, Topology 13 (1974); The theory of foliations of codimension greater than one, Comment. Math. Helv. 49 (1974); Existence of codimension-one foliations, Ann. of Math. 104 (1976).
A. Fathi, F. Laudenbach, V. Poénaru, Travaux de Thurston sur les surfaces, Astérisque 66–67 (1979).
W. P. Thurston, The Geometry and Topology of Three-Manifolds, Princeton lecture notes, from 1978; republished by the AMS in his collected works.
W. P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. AMS 6 (1982).
W. P. Thurston, Hyperbolic structures on 3-manifolds I: deformation of acylindrical manifolds, Ann. of Math. 124 (1986).
W. P. Thurston, A norm for the homology of 3-manifolds, Mem. AMS 59 (1986).
W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. AMS 19 (1988), 417–431.
J. Milnor and W. Thurston, On iterated maps of the interval, Lecture Notes in Math. 1342, Springer, 1988.
A. Douady and J. H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math. 171 (1993).
B. Rodin and D. Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987).
D. Sleator, R. Tarjan, W. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. AMS 1 (1988).
W. P. Thurston, Shapes of polyhedra and triangulations of the sphere, Geometry & Topology Monographs 1 (1998).
W. P. Thurston, On proof and progress in mathematics, Bull. AMS 30 (1994), 161–177.
W. P. Thurston, Three-Dimensional Geometry and Topology, Vol. 1, ed. S. Levy, Princeton University Press, 1997.
S. Friedl, A survey of the Thurston norm, arXiv:2108.08645.