Dennis Sullivan

How to use this course

Dennis Sullivan has changed fields several times, and each time the new field changed with him. In the 1960s he helped classify high-dimensional manifolds. In the 1970s he built homotopy theory one prime at a time and gave rational homotopy theory its working form through differential forms. From the late 1970s he turned to Kleinian groups and the iteration of rational maps, where his methods opened what is now called conformal dynamics. In 1999, with Moira Chas, he started string topology. The Abel Prize committee in 2022 summed this up as topology in its broadest sense.

Each technical lesson opens with a paragraph in italics for the general reader. Precise statements follow in boxes, then proof sketches and exercises with folded solutions. Lessons 04 to 06 belong to algebraic and geometric topology; Lessons 07 to 09 to geometry and dynamics; Lesson 10 returns to topology through loop spaces. The lessons can be read in any order after Lesson 00.

Timeline

Michigan, Houston and Rice, 1941–1963

Dennis Parnell Sullivan was born on 12 February 1941 in Port Huron, Michigan. The family soon moved to Houston, and he grew up in Texas, a state he still regards as home. By his own account he was not a diligent high-school student, and he was told not to bother applying to Rice University. He applied anyway and was admitted.

He entered Rice to study chemistry, aiming at the chemical industry that dominated the Houston economy. In his second year a mathematics course taught by Guy Johnson changed his direction. Sullivan has described a lecture in which the uniformization theorem appeared in a special case: any surface shaped like a sphere, however wrinkled, can be mapped onto the round sphere so that angles are preserved. The idea that such a statement could be proved, and proved for every shape at once, struck him as far more compelling than laboratory science. He switched to mathematics and graduated with a BA in 1963.

The episode foreshadows his whole career. Conformal maps, the maps that preserve angles, return in Lessons 07 to 09 as the central tool of his work in dynamics.

Princeton and the Hauptvermutung, 1963–1967

Sullivan went to Princeton for graduate study. His advisor was William Browder, one of the founders of surgery theory, the set of techniques that reduce the classification of manifolds in dimension five and above to algebra and homotopy theory. Browder in the United States and Sergei Novikov in the Soviet Union had shown how to decide whether a space with the homotopy type of a manifold actually is one. The next question was how many manifolds share a given homotopy type.

Sullivan’s thesis, Triangulating Homotopy Equivalences (January 1966), attacked this question in the piecewise-linear category. He found that the answer is governed by the homotopy type of a single classifying space, \(G/PL\), and computed enough of that space to make the theory usable. A 1967 announcement in the Bulletin of the American Mathematical Society applied the method to the Hauptvermutung, the old conjecture that homeomorphic triangulated manifolds are piecewise-linearly homeomorphic. Sullivan proved it under a mild homological condition in high dimensions. The work earned him the Oswald Veblen Prize in Geometry in 1971. Lesson 04 gives the details.

Warwick, Berkeley, MIT, 1966–1973

After the doctorate Sullivan spent 1966 at the University of Warwick as a NATO Fellow, where Christopher Zeeman had built a lively topology centre, and taught for a year at Princeton. From 1967 to 1969 he was a Miller Research Fellow at Berkeley. There he worked on the Adams conjecture in K-theory and learned the étale homotopy theory of algebraic varieties, which let him see Galois groups acting on topological spaces.

In 1969 he moved to MIT as a Sloan Fellow. His lecture notes of 1970, Geometric Topology, Part I: Localization, Periodicity and Galois Symmetry, were never formally published at the time, yet they circulated in photocopies for decades and shaped a generation of homotopy theorists. They introduced the localization and completion of spaces at sets of primes, recast the classification of manifolds in those terms, and stated what became the Sullivan conjecture on maps out of classifying spaces of finite groups. Andrew Ranicki edited them for Springer in 2005.

Sullivan was an invited speaker at the 1970 International Congress in Nice. By 1973 he was ready for a change of scene, and he spent 1973–74 at Orsay before taking up a permanent professorship at the IHÉS in 1974.

Surgery and triangulations

For the general reader. Two shapes can be indistinguishable by every test that allows continuous deformation, yet still be different manifolds. Surgery theory measures the gap. Sullivan showed that the measurement is controlled by one remarkable space, and computed it.

The structure set

Let \(M\) be a closed PL manifold of dimension \(n\). A homotopy triangulation of \(M\) is a simple homotopy equivalence \(f\colon N\to M\) from another closed PL manifold. Two are equivalent if they differ by a PL homeomorphism, up to homotopy. The set of equivalence classes is the structure set \(\mathcal S^{PL}(M)\). It answers the question: how many PL manifolds have the homotopy type of \(M\)?

Each homotopy triangulation determines a normal invariant, a homotopy class of maps \(M\to G/PL\). Here \(G\) is the monoid of stable self-homotopy equivalences of spheres and \(PL\) the stable PL homeomorphism group; \(G/PL\) classifies PL bundles together with a fibre homotopy trivialization.

The homotopy type of \(G/PL\)

Part (c) means that at 2 a normal invariant is recorded by characteristic classes, which Sullivan read geometrically as surgery obstructions along submanifolds of \(M\): his characteristic variety theorem. Part (b) gives a \(KO\)-theory orientation of PL bundles away from 2, the start of the Galois symmetry story of Lesson 05.

The Hauptvermutung

Idea. A homeomorphism is a homotopy triangulation whose normal invariant is topologically trivial. Sullivan showed the possible PL normal invariants of such maps are detected by the 2-torsion in \(H^3(M;\mathbb Z)\); when there is none, the normal invariant vanishes, and the surgery sequence with simple connectivity gives a PL homeomorphism. Andrew Casson obtained related results at the same time. In 1969 Robion Kirby and Laurence Siebenmann showed that the hypothesis is needed: for a closed manifold \(M\) of dimension at least 5, PL structures, when one exists, are classified up to isotopy by \(H^3(M;\mathbb Z/2)\), and there are homeomorphic PL manifolds that are not PL homeomorphic.

Localization and Galois symmetry

For the general reader. A number can be studied one prime at a time: its power of 2, its power of 3, and so on. Sullivan showed that whole spaces can be broken up the same way, then reassembled. Seen one prime at a time, familiar spaces acquire hidden symmetries that come from number theory.

Localization

For a set of primes \(P\), let \(\mathbb Z_P\) denote the integers with every prime outside \(P\) inverted. So \(\mathbb Z_{\{p\}}=\mathbb Z_{(p)}\) and \(\mathbb Z_{\varnothing}=\mathbb Q\).

The theorem extends to nilpotent spaces. For spheres one builds \(S^n_P\) as a mapping telescope of self-maps of degree \(k\) for \(k\) prime to \(P\). Taking \(P=\varnothing\) gives the rationalization \(X_{\mathbb Q}\), the subject of Lesson 06. Sullivan also defined the profinite completion \(\hat X\) and the \(p\)-adic completions \(\hat X_p\).

This is the topological version of the fact that \(\mathbb Z\) is the pullback of \(\mathbb Q\) and \(\hat{\mathbb Z}=\prod_p\mathbb Z_p\) over the finite adèles.

Galois symmetry and the Adams conjecture

Using the étale homotopy type of Michael Artin and Barry Mazur, Sullivan observed that the profinite completion of the complex points of an algebraic variety defined over \(\mathbb Q\) carries an action of \(\mathrm{Gal}(\bar{\mathbb Q}/\mathbb Q)\), which is invisible in ordinary topology. For the classifying spaces of vector bundles this action, through its abelianization \(\hat{\mathbb Z}^\times\), is given by the Adams operations \(\psi^k\).

Daniel Quillen gave the first proof, using Brauer lifting and finite fields. Sullivan’s proof uses the Galois action: a Galois symmetry of the completed bundle is a fibre homotopy equivalence, and the Galois element acting as \(k\) acts on \(K\)-theory as \(\psi^k\). The conjecture determines the image of the \(J\)-homomorphism in the stable homotopy groups of spheres.

Rational homotopy and minimal models

For the general reader. If we forget torsion, homotopy theory becomes linear algebra. Daniel Quillen proved this in 1969 with heavy machinery. Sullivan found a direct route through differential forms, the calculus of a manifold, and turned the subject into something one could compute by hand.

Polynomial forms

De Rham’s theorem says that differential forms on a manifold compute its real cohomology, multiplicatively. Sullivan wanted the same over \(\mathbb Q\) and for every space. He defined \(A_{PL}(X)\): on each singular simplex a polynomial differential form with rational coefficients, compatible with faces and degeneracies. It is a commutative differential graded algebra (cdga), and \(H^*(A_{PL}(X))\cong H^*(X;\mathbb Q)\) as algebras.

Minimal models

A Sullivan algebra is a cdga \((\Lambda V,d)\), free graded-commutative on a graded vector space \(V=\bigoplus_{n\ge1}V^n\), such that \(V\) has an exhausting filtration \(V(0)\subset V(1)\subset\cdots\) with \(d(V(0))=0\) and \(d(V(k))\subset\Lambda V(k-1)\). It is minimal if \(d(V)\subset\Lambda^{\ge2}V\), that is, the differential has no linear part. A minimal model of \(X\) is a minimal Sullivan algebra with a cdga map \((\Lambda V,d)\to A_{PL}(X)\) inducing an isomorphism in cohomology.

Construction. One builds \(V\) degree by degree: add closed generators to hit the cohomology not yet reached, then add generators whose differentials kill cohomology classes that are created but should not be there. The map \(\pi_n(X)\otimes\mathbb Q\to(V^n)^*\) comes from comparing this stepwise construction with the Postnikov tower of \(X_{\mathbb Q}\): each new generator of degree \(n\) is a stage \(K(\mathbb Q,n)\), and the differential is its \(k\)-invariant.

Formality

A space is formal if its minimal model is quasi-isomorphic to its cohomology algebra with zero differential. Then the cohomology ring determines the rational homotopy type. Higher Massey products are obstructions: on a formal space they all vanish.

The proof uses the \(dd^c\)-lemma of Hodge theory: on a compact Kähler manifold, a form that is \(d\)-closed, \(d^c\)-closed and \(d\)-exact is \(dd^c\)-exact. This yields quasi-isomorphisms \((\Omega^*,d)\leftarrow(\ker d^c,d)\to(H^*_{d^c}(M),0)\), and the last algebra has zero differential and the same cohomology. Pierre Deligne, Phillip Griffiths and John Morgan were Sullivan’s coauthors.

Kleinian groups and conformal measures

For the general reader. A Kleinian group is a discrete group of symmetries of hyperbolic space. Its orbits pile up on a fractal set at infinity, the limit set. Sullivan built a natural measure on that fractal and used it to compute its dimension, then proved a rigidity theorem that later became the template for his work on rational maps.

Critical exponent

Let \(\Gamma\) be a discrete group of isometries of hyperbolic space \(\mathbb H^{n+1}\), with limit set \(\Lambda(\Gamma)\subset S^n=\partial\mathbb H^{n+1}\). Fix a point \(o\). The Poincaré series \(\sum_{\gamma\in\Gamma}e^{-s\,d(o,\gamma o)}\) converges for large \(s\); the critical exponent is

Samuel Patterson, for Fuchsian groups in 1976, constructed a measure \(\mu\) on \(\Lambda(\Gamma)\) as a weak limit of weighted orbit measures \(\sum_\gamma e^{-s\,d(o,\gamma o)}\delta_{\gamma o}\) as \(s\downarrow\delta\). Sullivan extended the construction to all dimensions and showed the measure is conformal of dimension \(\delta\): \(\gamma^*\mu=|\gamma'|^{\delta}\mu\) for all \(\gamma\in\Gamma\), with \(|\gamma'|\) the derivative in the spherical metric.

The convex cocompact case appeared in the 1979 paper; the geometrically finite case, with cusps, in 1984. Sullivan also linked \(\delta\) to spectral theory: for geometrically finite \(\Gamma\) with \(\delta>n/2\), the bottom of the spectrum of the Laplacian on \(\mathbb H^{n+1}/\Gamma\) is \(\lambda_0=\delta(n-\delta)\).

No invariant line fields

Invariant line fields are exactly what one needs to deform a group quasiconformally. The theorem says that any deformation of \(\Gamma\) is controlled by its action on the domain of discontinuity, the complement of the limit set. It underlies the rigidity results needed in William Thurston’s geometrization programme, and its analogue for rational maps remains a central open problem (Lesson 08). With Thurston, Sullivan also proposed the density conjecture, that geometrically finite groups are dense among all finitely generated Kleinian groups; it was proved by Ken’ichi Ohshika and by Hossein Namazi and Juan Souto around 2011–2012.

No wandering domains

For the general reader. Iterate a rational function on the complex plane. Some regions behave calmly: nearby points stay nearby. Pierre Fatou and Gaston Julia asked around 1920 whether such a calm region could keep moving to new places forever, never repeating. Sullivan proved in 1985 that it cannot, by importing a tool from Kleinian groups.

Setting

Let \(f\colon\hat{\mathbb C}\to\hat{\mathbb C}\) be a rational map of degree \(d\ge2\). The Fatou set \(F(f)\) is the set of points with a neighbourhood on which the iterates \(f^n\) form a normal family; the Julia set \(J(f)\) is its complement. Both are completely invariant. A component \(U\) of \(F(f)\) is wandering if the components \(f^n(U)\), \(n\ge0\), are pairwise distinct.

Combined with work of Fatou, Hubert Cremer, Carl Ludwig Siegel and Michael Herman, this gives a complete list. Every periodic Fatou component is one of: an attracting basin, a parabolic basin, a Siegel disc, or a Herman ring. Each attracting or parabolic cycle attracts a critical point, so there are at most \(2d-2\) of them.

Proof sketch

The key tool is the measurable Riemann mapping theorem of Charles Morrey, Lars Ahlfors and Lipman Bers: for any measurable \(\mu\) on \(\hat{\mathbb C}\) with \(\|\mu\|_\infty<1\) there is a quasiconformal homeomorphism \(\phi\) solving \(\bar\partial\phi=\mu\,\partial\phi\), unique after normalization and depending holomorphically on \(\mu\).

1. Suppose \(U\) wanders. After replacing \(U\) by a forward image one may assume no \(f^n(U)\) contains a critical point (only finitely many do).
2. Choose any Beltrami coefficient \(\mu\) on \(U\), and spread it over the grand orbit of \(U\) by pulling back and pushing forward with \(f\). Set \(\mu=0\) elsewhere. The result is \(f\)-invariant: \(f^*\mu=\mu\).
3. Solve for \(\phi_\mu\). Invariance makes \(f_\mu=\phi_\mu\circ f\circ\phi_\mu^{-1}\) conformal, hence a rational map of the same degree.
4. Since \(U\) wanders, one has infinitely many independent choices of \(\mu\) on \(U\), giving an infinite-dimensional family of maps \(f_\mu\). But rational maps of degree \(d\) form a space of dimension \(2d+1\), and conjugacy by Möbius maps removes 3 dimensions. Sullivan shows that enough of the deformations give genuinely different conjugacy classes, a contradiction.

The last step requires care, because many \(\mu\) give conjugate maps; Sullivan uses the structure of the boundary of \(U\) and the action on Teichmüller spaces of the grand orbit. The strategy is the dictionary in action: methods from Kleinian groups, where the analogous finiteness was Ahlfors’s finiteness theorem, transferred to rational maps. Sullivan’s dictionary between the two theories has guided the field since.

The theorem fails for transcendental entire functions: Irvine Noel Baker found entire maps with wandering domains. The parameter count in step 4 has no finite-dimensional analogue there.

Renormalization and universality

For the general reader. In 1975 Mitchell Feigenbaum noticed that very different systems pass from order to chaos through period doubling at the same geometric rate, about 4.669. Why should a fish population model and a fluid share a number? The explanation is a fixed point of a renormalization operator acting on maps. Sullivan gave the first conceptual proof that renormalization converges.

Period doubling

For the logistic family \(f_r(x)=rx(1-x)\) on \([0,1]\), let \(r_n\) be the parameter at which the attracting cycle of period \(2^{n-1}\) loses stability and a cycle of period \(2^n\) appears. Feigenbaum, and independently Pierre Coullet and Charles Tresser, observed that

and that the same \(\delta\) appears for every smooth unimodal family with a quadratic critical point.

Renormalization

A unimodal map \(g\) is renormalizable if, on a small interval \(I\) around its critical point, \(g^2\) maps \(I\) into itself and is again unimodal. Rescaling \(I\) to the original interval defines \(\mathcal R g\). The conjecture asserts that \(\mathcal R\) has a fixed point \(g^*\) (satisfying the Cvitanović–Feigenbaum equation \(g^*(x)=-\alpha\,g^*\big(g^*(-x/\alpha)\big)\) with \(\alpha\approx2.5029\)), that \(g^*\) is hyperbolic with a single unstable eigenvalue equal to \(\delta\), and that all infinitely renormalizable maps converge to it.

Sullivan’s proof combined “real a priori bounds” (uniform control of the geometry of the nested intervals) with complex methods: extending the maps to quadratic-like maps in the plane and using Teichmüller theory to contract distances. Oscar Lanford had earlier proved the existence of the fixed point with computer assistance (1982). McMullen proved that the convergence is exponentially fast for bounded type (1996), and Lyubich established the hyperbolicity of the renormalization horseshoe in full (1999), completing the conceptual explanation of universality.

String topology

For the general reader. Physicists study strings, loops moving through space. In 1999 Moira Chas and Sullivan found that the space of all loops on a manifold carries a rich algebra: two families of loops can be intersected where they meet and joined into a new family. The structure echoes those found in string theory and in the deformation theory of algebras.

The loop product

Let \(M\) be a closed oriented manifold of dimension \(d\) and \(LM=\operatorname{Map}(S^1,M)\) its free loop space, with evaluation \(ev\colon LM\to M\) at the base point. Given cycles \(a\in H_p(LM)\) and \(b\in H_q(LM)\), consider the family \(a\times b\) of pairs of loops. Intersect transversally with the condition \(ev(\alpha)=ev(\beta)\), a codimension-\(d\) condition in \(M\times M\); at each surviving pair, the two loops share a base point and can be concatenated. The result is a class \(a\bullet b\in H_{p+q-d}(LM)\).

Restricted to the constant loops \(M\subset LM\), the loop product is the classical intersection product on \(H_*(M)\). On the based loops \(\Omega M=ev^{-1}(x_0)\) it is related to the Pontryagin product. Ralph Cohen and John Jones gave a homotopy-theoretic construction using a Thom collapse and showed that \(\mathbb H_*(LM)\) is isomorphic, for simply connected \(M\), to the Hochschild cohomology of the cochain algebra of \(M\), connecting string topology with deformation theory. The string bracket on surfaces recovers the Goldman bracket on free homotopy classes of curves.

IHÉS, New York, Stony Brook

From 1974 Sullivan was a permanent professor at the Institut des Hautes Études Scientifiques at Bures-sur-Yvette, near Paris. He became one of its central figures, known for long open-ended discussions in which visitors explained their problems and Sullivan asked deceptively simple questions. In 1977 he invited Mikhail Gromov to the institute as a visitor. In those years he moved from topology to dynamics, working with David Ruelle on currents and flows, with William Parry on invariants of flows, and on foliations. His 1983 paper Conformal dynamical systems set out the analogies between Kleinian groups and rational maps.

In 1981 he accepted the Albert Einstein Chair in Science at the Graduate Center of the City University of New York, holding it jointly with a part-time IHÉS position. In New York he ran a long-running seminar on geometry and chaos, and supervised a stream of doctoral students in dynamics, among them Yunping Jiang, Edson de Faria and Adam Epstein. Curtis McMullen, who completed his Harvard thesis under Sullivan’s supervision in 1985, went on to win the Fields Medal in 1998.

In the mid-1990s he joined Stony Brook University and gave up his professorship at the IHÉS; he became a SUNY Distinguished Professor in 1998. At Stony Brook his later students worked in string topology and algebraic models, and he helped found the Simons Center for Geometry and Physics, serving on its board. With Moira Chas he started string topology in 1999. With Jim Simons he wrote on differential characters and structured bundles. Since around 2010 much of his effort has gone to the three-dimensional incompressible Euler and Navier–Stokes equations, seeking combinatorial and algebraic models for fluid motion. His Abel lecture in 2022 was devoted to these questions.

His work on quasiconformal and Lipschitz structures also deserves mention: he proved that topological manifolds of dimension other than four admit Lipschitz structures, unique up to isotopy, and with Simon Donaldson (1989) studied quasiconformal 4-manifolds, a line extended by Alain Connes, Sullivan and Nicolae Teleman (1994) to an index theorem for quasiconformal manifolds. With Burton Rodin (1987) he proved Thurston’s conjecture that circle packings converge to the Riemann map.

Honours and influence

Sullivan received the Oswald Veblen Prize in Geometry (1971), the Prix Élie Cartan of the French Academy of Sciences (1981), the King Faisal International Prize for Science (1994), the US National Medal of Science (2004), the AMS Steele Prize for Lifetime Achievement (2006), the Wolf Prize in Mathematics (2010, shared with Shing-Tung Yau, for algebraic topology and conformal dynamics), the Balzan Prize (2014) and the Abel Prize (2022). He was elected to the US National Academy of Sciences in 1983 and to the American Academy of Arts and Sciences in 1991, and served as vice-president of the American Mathematical Society from 1990 to 1993. He gave a plenary address at the International Congress of Mathematicians in Vancouver in 1974.

Influence

• Topology: the computation of \(G/PL\), localization and completion, and the Sullivan conjecture shaped geometric and algebraic topology. Minimal models remain the standard computational tool of rational homotopy theory.
• Complex geometry: formality of Kähler manifolds is one of the basic topological restrictions on Kähler manifolds.
• Dynamics: quasiconformal deformation, the dictionary with Kleinian groups, and renormalization theory created modern holomorphic dynamics; his students and their students carried it on.
• Geometry and topology of loops: string topology connects manifold topology with Hochschild cohomology, symplectic field theory and the Goldman–Turaev structures on surfaces.

The thread through all of it is a preference for structures that can be seen and computed: polynomial forms in place of abstract simplicial machinery, deformations in place of existence theorems, and families of loops in place of homotopy groups.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Dennis Parnell Sullivan, MacTutor History of Mathematics.
2. Dennis Sullivan, Wikipedia.
3. The Abel Prize, Dennis P. Sullivan, Abel Prize 2022, citation and lectures; biography (PDF).
4. Balzan Foundation, Dennis Sullivan: bio-bibliography.
5. Dennis Parnell Sullivan, Mathematics Genealogy Project.

Mathematics

1. D. Sullivan, On the Hauptvermutung for manifolds, Bull. AMS 73 (1967).
2. D. Sullivan, Geometric Topology: Localization, Periodicity and Galois Symmetry (the 1970 MIT notes), ed. A. Ranicki, K-Monographs in Mathematics 8, Springer, 2005.
3. D. Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. of Math. 100 (1974).
4. P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975).
5. D. Sullivan, Infinitesimal computations in topology, Publ. Math. IHÉS 47 (1977).
6. D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Publ. Math. IHÉS 50 (1979).
7. D. Sullivan, Quasiconformal homeomorphisms and dynamics I: solution of the Fatou–Julia problem on wandering domains, Ann. of Math. 122 (1985).
8. B. Rodin, D. Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987).
9. M. Chas, D. Sullivan, String topology, arXiv:math/9911159.
10. Y. Félix, S. Halperin, J.-C. Thomas, Rational Homotopy Theory, Springer GTM 205, 2001.
11. A. Ranicki, The Hauptvermutung Book, Kluwer, 1996.
12. C. McMullen, Complex Dynamics and Renormalization, Annals of Mathematics Studies 135, Princeton, 1994.
13. Rational homotopy theory and String topology, Wikipedia.