John Milnor

How to use this course

John Milnor’s 1956 discovery of exotic spheres created differential topology as a separate subject. Over the next seven decades he moved from knots to smooth manifolds, then to singularities, algebra and finally dynamical systems. In each field he proved a defining theorem, posed questions that occupied others for decades, and wrote a short book that became the standard introduction. His honours include the Fields Medal, the Wolf Prize, the Abel Prize and all three of the American Mathematical Society’s Steele Prizes.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow his work in rough chronological order: knots (1950–54), differential topology (1956–65), singularities (1968), algebra (1968–71) and dynamics (from the late 1970s). Milnor is living; Part III covers his career and public honours only.

Timeline

Orange and Princeton, 1931–1951

John Willard Milnor was born in Orange, New Jersey, on 20 February 1931. He entered Princeton University as an undergraduate and quickly stood out. He was a Putnam Fellow, one of the top scorers in the national Putnam competition, in both 1949 and 1950.

As an undergraduate he took up a question about knots in space posed by Karol Borsuk: how much must a closed curve bend in order to be knotted? Werner Fenchel had shown that every closed curve has total curvature at least \(2\pi\). Milnor proved that a knotted curve needs more than \(4\pi\). The paper, On the total curvature of knots, appeared in the Annals of Mathematics in 1950. István Fáry had found the same result independently, and it is now the Fáry–Milnor theorem (Lesson 04). Milnor received his AB in 1951.

Links and the thesis, 1951–1954

Milnor stayed at Princeton for graduate work under Ralph Fox, a knot theorist who was developing the free differential calculus. His thesis, on the isotopy of links, was completed in 1954. In the paper Link groups (1954) he introduced numerical invariants of links now called Milnor’s \(\bar\mu\)-invariants. They are computed from the lower central series of the link group and detect linking that pairwise linking numbers miss, as in the Borromean rings, where each pair of components is unlinked but the three together are not.

In the same years he wrote papers on game theory, including Sums of positional games (1953), and on the algebraic topology of classifying spaces. His construction of a universal bundle for any topological group, by an infinite join, appeared in 1956 and is still the standard one.

The young professor, 1953–1962

Princeton appointed Milnor an instructor in 1953, before he had finished his doctorate. He held an Alfred P. Sloan Fellowship from 1955 to 1959, became a full professor in 1960 and was given the Henry Putnam chair in 1962.

The decade was extraordinarily productive. In 1956 he constructed smooth manifolds that are homeomorphic but not diffeomorphic to the 7-sphere (Lesson 05). In 1958 he computed the structure of the dual of the Steenrod algebra, a result central to stable homotopy theory, and with John Moore he later wrote the basic paper on Hopf algebras. In 1961 he disproved the Hauptvermutung, the conjecture that two homeomorphic simplicial complexes have isomorphic subdivisions, using the Reidemeister torsion of lens spaces. His lecture notes from these years, on Morse theory, characteristic classes and the h-cobordism theorem, circulated widely before they became books.

In 1962 he received the Fields Medal at the International Congress in Stockholm, for his work in differential topology, and gave an invited address at the same Congress. He began a long period as an editor of the Annals of Mathematics that year and was elected to the National Academy of Sciences in 1963.

The total curvature of knots

For the general reader. Imagine walking along a closed loop of wire and adding up how much your direction turns. A flat circle turns once, a total of \(2\pi\). Milnor proved, as an undergraduate, that if the loop is knotted you must turn by more than twice that. The proof averages over all directions from which you can look at the wire.

Total curvature

Let \(\gamma:\mathbb R/L\mathbb Z\to\mathbb R^3\) be a smooth closed curve parametrized by arc length, with unit tangent \(T(s)=\gamma'(s)\) and curvature \(\kappa(s)=|T'(s)|\). The total curvature is \(K(\gamma)=\int_0^L\kappa\,ds\), the length of the tangent indicatrix \(T\) on the unit sphere \(S^2\).

The averaging argument

For a unit vector \(u\), consider the height function \(h_u(s)=\langle\gamma(s),u\rangle\). Its critical points are the \(s\) with \(T(s)\perp u\), that is, the points where the curve \(T\) meets the great circle \(u^\perp\). For almost every \(u\) these are non-degenerate and alternate between maxima and minima; let \(\mu(u)\) be the number of local maxima. Then \(T\) meets \(u^\perp\) in \(2\mu(u)\) points. The Crofton formula on \(S^2\) says the length of a curve equals \(\pi\) times the average number of its intersections with a random great circle (Exercise 4.1). Hence

Since every height function on a closed curve has at least one maximum, \(K\ge2\pi\). Now suppose \(\mu(u)=1\) for some \(u\). Then each horizontal plane \(h_u=t\) between the minimum and maximum meets \(\gamma\) in exactly two points. Joining these pairs by straight segments sweeps out an embedded disc bounded by \(\gamma\), so \(\gamma\) is unknotted. A knotted curve therefore has \(\mu(u)\ge2\) for almost every \(u\), and \(K\ge4\pi\). An additional argument, using that \(\mu\) is locally constant on an open set of directions where the inequality can be improved, rules out equality.

Exotic 7-spheres

For the general reader. Topology studies shapes up to continuous deformation; differential topology asks also that the shapes be smooth, so that calculus works on them. Before 1956 it was assumed that a shape topologically equal to a sphere must be smoothly equal to it too. Milnor found 7-dimensional spaces that are spheres topologically but cannot be smoothed into the standard sphere. The discovery showed that smooth structure is a genuinely new layer of geometry.

The construction

Oriented \(S^3\)-bundles over \(S^4\) with structure group \(\mathrm{SO}(4)\) are classified by \(\pi_3(\mathrm{SO}(4))\cong\mathbb Z\oplus\mathbb Z\). Identifying \(\mathbb R^4\) with the quaternions, the pair \((h,j)\) corresponds to the clutching map \(f_{h,j}(q)\cdot v=q^h v q^j\) for unit quaternions \(q\). Let \(\xi_{h,j}\) be the resulting bundle. Its Euler class is \(e=h+j\) and its first Pontryagin class is \(p_1=\pm2(h-j)\), times a generator of \(H^4(S^4)\).

Take \(h+j=1\) and write \(k=h-j\), an odd integer. The total space \(M^7_k\) of the sphere bundle is a smooth closed 7-manifold. Milnor found an explicit function on \(M_k\) with exactly two critical points, both non-degenerate. By a theorem of Georges Reeb, \(M_k\) is then homeomorphic to \(S^7\).

The invariant

Proof. Given two such \(W\), glue them along \(M\) with one orientation reversed to get a closed smooth 8-manifold \(N\). Hirzebruch’s signature theorem gives \(\sigma(N)=\frac{7p_2-p_1^2}{45}[N]\), and \(p_2[N]\) is an integer. So \(45\sigma(N)+p_1^2[N]\equiv0\pmod 7\), that is \(3\sigma+p_1^2\equiv0\), and multiplying by 2, \(2p_1^2-\sigma\equiv0\pmod7\). Both terms are additive under the gluing, so the two choices of \(W\) give the same value. If \(M\cong S^7\), glue in a disc \(D^8\) to make \(W\) closed, and the same congruence gives \(\lambda=0\).

Take \(W_k\) to be the associated \(D^4\)-disc bundle. Its intersection form on \(H^4\cong\mathbb Z\) is \((e)=(1)\), so \(\sigma=1\), and \(p_1^2[W_k]=(2k)^2=4k^2\). Then \(\lambda=8k^2-1\equiv k^2-1\pmod 7\). This is non-zero unless \(k\equiv\pm1\pmod 7\).

Morse theory and the h-cobordism theorem

For the general reader. Flood a landscape slowly and watch the shape of the water change. It changes only when the water level passes a peak, a pass or a pit, and each kind of point changes it in a standard way. Morse theory turns this picture into a way of building any smooth manifold from simple pieces. Milnor’s two short books on the subject made it the working tool of a generation of topologists.

Morse functions

A smooth function \(f:M\to\mathbb R\) on a manifold is a Morse function if its critical points are non-degenerate: the Hessian at each critical point is invertible. The index of a critical point is the number of negative eigenvalues of the Hessian. By the Morse lemma, near a critical point of index \(\lambda\) there are coordinates with \(f=f(p)-x_1^2-\cdots-x_\lambda^2+x_{\lambda+1}^2+\cdots+x_n^2\).

Consequently a closed manifold with a Morse function has the homotopy type of a CW complex with one \(\lambda\)-cell for each critical point of index \(\lambda\). With \(C_\lambda\) the number of such critical points and \(b_\lambda\) the Betti numbers over a field, this gives the Morse inequalities \(C_\lambda\ge b_\lambda\) and \(\sum(-1)^\lambda C_\lambda=\chi(M)\). In the last part of his book Milnor gave Bott’s proof of the periodicity theorem by applying Morse theory to the energy function on the space of paths in a Lie group.

The h-cobordism theorem

A cobordism \((W;M_0,M_1)\) is a compact manifold with boundary \(M_0\sqcup M_1\). It is an h-cobordism if both inclusions \(M_i\hookrightarrow W\) are homotopy equivalences.

The proof chooses a Morse function on \(W\) with \(f^{-1}(0)=M_0\), \(f^{-1}(1)=M_1\), and cancels its critical points in pairs. Two critical points of adjacent index can be cancelled when the descending sphere of one meets the ascending sphere of the other transversely in a single point. The Whitney trick, which needs dimension at least 5 for the level sets, moves the spheres to achieve this. Milnor’s Lectures on the h-cobordism theorem (1965), written with L. Siebenmann and J. Sondow, gave the proof in the form now standard. A homotopy \(n\)-sphere with \(n\ge6\) minus two discs is an h-cobordism between two \((n-1)\)-spheres, which leads to Smale’s proof of the Poincaré conjecture in dimensions \(\ge5\): every homotopy \(n\)-sphere is homeomorphic to \(S^n\). It is generally not diffeomorphic to \(S^n\), as Lesson 05 showed.

Groups of homotopy spheres

For the general reader. After finding exotic 7-spheres, Milnor and Michel Kervaire counted them, in every dimension. The answer links the smooth structures on spheres to the homotopy groups of spheres and to Bernoulli numbers. In dimension 7 there are exactly 28.

The group \(\Theta_n\)

Let \(\Theta_n\) be the set of h-cobordism classes of oriented smooth homotopy \(n\)-spheres. Connected sum makes it an abelian group with identity \(S^n\) and inverse given by reversing orientation. For \(n\ge5\) h-cobordism is the same as diffeomorphism (Lesson 06), so \(\Theta_n\) is the group of smooth structures on \(S^n\). Let \(bP_{n+1}\subseteq\Theta_n\) be the subgroup of homotopy spheres that bound parallelizable manifolds.

Ideas. Every homotopy sphere is stably parallelizable, so it has a framing, defining an element of \(\pi_n^s\) via the Pontryagin–Thom construction, well defined modulo the image of \(J\). If this element vanishes, \(\Sigma\) bounds a parallelizable manifold \(W\). Surgery on \(W\) tries to kill its homology; the only obstruction is in the middle dimension. For \(\dim W=4m\) the obstruction is the signature, which must be divisible by 8, and the order of \(bP_{4m}\) comes from the smallest signature of an almost-parallelizable closed \(4m\)-manifold, computed with the \(\hat A\)-genus and Adams’ results on \(J\). For \(\dim W\equiv2\pmod 4\) the obstruction is the Kervaire invariant in \(\mathbb Z/2\). The determination of when it is non-zero was completed by Hill, Hopkins and Ravenel in 2009 (except dimension 126, for which a solution was announced in 2024).

Combinatorial structures

Milnor also showed that topology, smooth structure and piecewise-linear structure come apart in other ways. In 1961 he disproved the Hauptvermutung with two simplicial complexes, built from lens spaces, that are homeomorphic but have no common subdivision; Reidemeister torsion tells them apart. His 1966 survey Whitehead torsion set out the torsion invariants used in the s-cobordism theorem.

Singular points and the Milnor fibration

For the general reader. Where a complex polynomial surface has a singular point, cut out a tiny sphere around it. The intersection is a knot-like manifold, and the surrounding space winds around it in an orderly way, like the pages of an open book around its spine. Milnor’s 1968 book proved this and found the numbers that measure how complicated the singular point is. Some of these knots are exotic spheres.

The closure of each fibre is a compact manifold with boundary \(K\), so \(K\) is a fibred link and the fibres are its Seifert surfaces. The monodromy of the fibration acts on \(H_n(\text{fibre})\cong\mathbb Z^\mu\), and its characteristic polynomial \(\Delta(t)\) determines the topology of \(K\) in many cases: for \(n\ne2\), \(K\) is a topological sphere iff \(\Delta(1)=\pm1\).

Brieskorn spheres

For \(f=z_0^{a_0}+\cdots+z_n^{a_n}\) the monodromy can be computed explicitly (Brieskorn and Pham). Egbert Brieskorn showed in 1966 that the links

realize all 28 homotopy 7-spheres. So the exotic spheres of Lesson 05 appear naturally as links of algebraic singularities, a connection Milnor’s book made central.

Growth of groups and Milnor K-theory

For the general reader. Around 1968–1970 Milnor turned to algebra twice. First, he asked how fast a group grows when you count the elements you can write as words of a given length, and he tied the answer to the curvature of spaces. Second, he defined a simple algebraic object attached to any field and conjectured that it captures the theory of quadratic forms. Both questions grew into large fields.

Growth

For a group \(G\) with finite symmetric generating set \(S\), let \(\beta(r)\) be the number of elements of word length at most \(r\). Changing \(S\) changes \(\beta\) only up to linear rescaling of \(r\), so polynomial or exponential growth is a property of \(G\).

Part (b) follows from (a) by comparing word balls with volumes of balls in the universal cover, which grow exponentially in negative curvature and at most polynomially when \(\mathrm{Ric}\ge0\) (Bishop’s inequality). In separate 1968 papers, Milnor and Joseph Wolf showed that a finitely generated solvable group has either polynomial or exponential growth. Milnor asked whether intermediate growth can occur at all. Rostislav Grigorchuk constructed a group of intermediate growth in 1984. Mikhail Gromov proved in 1981 that polynomial growth implies virtually nilpotent, a converse to results of Milnor and Wolf.

Milnor K-theory

For a field \(F\), Milnor defined in 1970

written additively with symbols \(\{a_1,\dots,a_n\}\). So \(K^M_1(F)=F^\times\) and \(K^M_2(F)\) agrees with Quillen’s \(K_2(F)\) by Matsumoto’s theorem.

Vladimir Voevodsky proved the first isomorphism using motivic cohomology, work recognized by his 2002 Fields Medal; the second was completed by Orlov, Vishik and Voevodsky in 2007. The generalization to odd primes, the Bloch–Kato conjecture, was proved by Voevodsky with contributions from Markus Rost and others.

Dynamics: kneading and the Mandelbrot set

For the general reader. Repeat a simple rule such as \(x\mapsto x^2+c\) again and again, and the sequence of values can settle down, cycle or behave chaotically. From the mid-1970s Milnor made this subject his own. With William Thurston he found a combinatorial code that captures the behaviour of maps of an interval, and his textbook on complex dynamics became the standard entry to the theory of Julia and Mandelbrot sets.

Kneading theory

A map \(f:[a,b]\to[a,b]\) is piecewise monotone if the interval splits into finitely many laps on which \(f\) is monotone. The kneading sequence records, for the orbit of each turning point, on which lap each iterate lands. Milnor and Thurston packaged this information in a formal power series, the kneading determinant, whose smallest zero determines the topological entropy \(h(f)\), the exponential growth rate of the number of laps of \(f^n\).

Monotonicity says that as \(c\) decreases from \(\tfrac14\) toward \(-2\), periodic orbits are created and never destroyed. It was one of the first deep results about the full quadratic family, and its proof used complex methods, an early sign of the interplay between real and complex dynamics.

Julia and Mandelbrot sets

For a polynomial \(p\) of degree \(d\ge2\), the filled Julia set \(K(p)\) is the set of \(z\in\mathbb C\) with bounded orbit, and the Julia set \(J(p)=\partial K(p)\) is where the dynamics is chaotic. For \(p_c(z)=z^2+c\), Pierre Fatou and Gaston Julia showed that \(K(p_c)\) is connected iff the critical orbit \(0\mapsto c\mapsto c^2+c\mapsto\cdots\) is bounded. The Mandelbrot set is

Douady and Hubbard proved that \(\mathcal M\) is connected; whether it is locally connected (MLC) is open and implies density of hyperbolic dynamics in the quadratic family. Milnor’s Dynamics in One Complex Variable (1999, third edition 2006) grew from his Stony Brook lectures, and his papers on parameter spaces of cubic polynomials, on self-similarity of \(\mathcal M\), and on the geometry of Lattès maps shaped the field.

Institute and Stony Brook, 1967–present

Milnor received the National Medal of Science in 1967. That year he left Princeton University, and after short appointments at UCLA and MIT he joined the faculty of the Institute for Advanced Study in 1970, where he stayed for two decades. He served as vice-president of the American Mathematical Society in 1975–76. His interests moved steadily from topology toward dynamical systems, first on the interval and then in the complex plane, an area he helped revive alongside Adrien Douady, John Hubbard, Dennis Sullivan and Thurston.

In 1989 he became the first director of the Institute for Mathematical Sciences at Stony Brook University, a centre focused on dynamics and geometry. He later served as its co-director, and holds the title of Distinguished Professor at Stony Brook. That year he also received the Wolf Prize in Mathematics.

His doctoral students at Princeton included John Mather, Laurent Siebenmann, Michael Spivak, Jon Folkman, Tadatoshi Akiba and Jonathan Sondow. Many more mathematicians count him as a mentor through his lectures and books.

Expositor and legacy

Milnor’s books are short, precise and readable, and several have been in continuous use for more than half a century: Morse Theory (1963), Topology from the Differentiable Viewpoint (1965), Lectures on the h-Cobordism Theorem (1965), Singular Points of Complex Hypersurfaces (1968), Introduction to Algebraic K-Theory (1971), Symmetric Bilinear Forms with Dale Husemoller (1973), Characteristic Classes with James Stasheff (1974) and Dynamics in One Complex Variable (1999). The American Mathematical Society recognized this with its 2004 Steele Prize for exposition.

He also produced memorable single results outside his main lines. In 1964 he found two flat tori of dimension 16 that have the same Laplace spectrum but are not isometric, answering negatively, in high dimension, the question later popularized as whether one can hear the shape of a drum.

In 2011 the Norwegian Academy of Science and Letters awarded him the Abel Prize for pioneering discoveries in topology, geometry and algebra. In the same year he received the Steele Prize for lifetime achievement, completing the set after the 1982 prize for his paper on the 7-sphere. In 2020 the Russian Academy of Sciences awarded him its Lomonosov Gold Medal.

Legacy

• Differential topology: exotic spheres, the Kervaire–Milnor computation and the h-cobordism methods founded the classification of high-dimensional manifolds and surgery theory.
• Singularities: the Milnor fibration and Milnor number are the basic invariants of isolated hypersurface singularities.
• Algebra: Milnor K-theory and the Milnor conjecture led to motivic cohomology; his questions on growth started geometric group theory.
• Dynamics: kneading theory and his work on polynomial parameter spaces are central in one-dimensional and holomorphic dynamics.
• Honours: Fields Medal (1962), National Medal of Science (1967), Steele Prizes (1982, 2004, 2011), Wolf Prize (1989), Abel Prize (2011), Lomonosov Gold Medal (2020); member of the National Academy of Sciences, the American Academy of Arts and Sciences and the American Philosophical Society, and foreign member of the Russian Academy of Sciences.

Connections

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

Teachers and advisors

• Ralph Fox 1954

Students

• John Mather
• Jon Folkman
• Jonathan Sondow
• Laurent Siebenmann
• Michael Spivak
• Tadatoshi Akiba

Influenced by

• Friedrich Hirzebruch

Influenced

• Mikhail Gromov
• Vladimir Voevodsky

Colleagues and collaborators

• Dennis Sullivan
• William Thurston
• Dale Husemoller 1973
• Egbert Brieskorn
• István Fáry
• James Stasheff 1974
• John C. Moore
• Michel Kervaire 1963
• Stephen Smale

Family

• Dusa McDuff

Also mentioned in this life

• Leonhard Euler
• Évariste Galois
• Pierre-Simon Laplace
• Sophus Lie
• Henri Poincaré
• Gregorio Ricci-Curbastro

Fields

• knot theory 3 lives
• dynamical systems 7 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, John Willard Milnor, MacTutor History of Mathematics.
2. John Milnor, Wikipedia.
3. Norwegian Academy of Science and Letters, Biography of John Milnor, Abel Prize 2011.
4. M. Raussen and C. Skau, Interview with John Milnor, Notices of the AMS 59 (2012).

Mathematics

1. J. Milnor, On the total curvature of knots, Ann. of Math. 52 (1950).
2. J. Milnor, Link groups, Ann. of Math. 59 (1954).
3. J. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math. 64 (1956).
4. J. Milnor, Two complexes which are homeomorphic but combinatorially distinct, Ann. of Math. 74 (1961).
5. M. Kervaire and J. Milnor, Groups of homotopy spheres I, Ann. of Math. 77 (1963).
6. J. Milnor, Morse Theory, Princeton, 1963.
7. J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. 51 (1964).
8. J. Milnor, Lectures on the h-Cobordism Theorem, Princeton, 1965.
9. J. Milnor, Whitehead torsion, Bull. AMS 72 (1966).
10. J. Milnor, A note on curvature and fundamental group, J. Differential Geom. 2 (1968).
11. J. Milnor, Singular Points of Complex Hypersurfaces, Princeton, 1968.
12. J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1970).
13. J. Milnor and J. Stasheff, Characteristic Classes, Princeton, 1974.
14. J. Milnor and W. Thurston, On iterated maps of the interval, in Dynamical Systems, Lecture Notes in Math. 1342, Springer, 1988.
15. J. Milnor, Dynamics in One Complex Variable, 3rd ed., Princeton, 2006. Early version, arXiv:math/9201272.
16. J. Milnor, Differential topology forty-six years later, Notices of the AMS 58 (2011).