Solomon Lefschetz

How to use this course

Solomon Lefschetz had three mathematical careers. In the first, in Kansas, he used topology to study algebraic surfaces and varieties, and his results still frame how geometers think about the shape of a complex variety. In the second, at Princeton, he turned topology itself into an algebraic subject, with the fixed-point theorem, duality and intersection theory, and gave it the name algebraic topology. In the third, begun when he was nearly sixty, he built an American school of nonlinear differential equations and control theory. He also edited the Annals of Mathematics for thirty years and helped create the Mexican school of mathematics.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04 to 06 form a sequence on the topology of algebraic varieties. Lessons 07 to 09 form a short sequence on fixed points, and Lesson 10 needs nothing from the others.

Timeline

Moscow and Paris, 1884–1905

Solomon Lefschetz was born in Moscow on 3 September 1884. His parents, Alexander Lefschetz and Sarah (or Vera) Lifschitz, were Jewish merchants who held Turkish citizenship. Soon after his birth the family moved to Paris, where the children were educated. French became his first language, and he kept a French accent and a French sense of mathematical style all his life.

He entered the École Centrale des Arts et Manufactures in 1902. It trained engineers, not mathematicians, but it gave a solid grounding in analysis and mechanics. He graduated in 1905 as a mechanical engineer, one of the youngest in a large class. A career in French academic mathematics, which passed through the École Normale Supérieure and the agrégation, was not open to a foreign engineering graduate. He decided to try his fortune in the United States.

Pittsburgh and Worcester, 1905–1911

Lefschetz arrived in the United States in November 1905. He worked first at the Baldwin Locomotive Works near Philadelphia, and in January 1907 joined the Westinghouse Electric and Manufacturing Company in Pittsburgh as an engineering apprentice. In November 1907 a transformer exploded during a test. He lost both hands and forearms. He was 23.

After his recovery he stayed with Westinghouse for three more years and turned to mathematics. In the autumn of 1910 he left the company to enrol as a graduate student at Clark University in Worcester, Massachusetts, on a small fellowship. Clark was a small research university founded on the German model, and its mathematicians included William Edward Story, who worked in algebraic geometry and invariant theory.

Story suggested a problem: what is the largest number of cusps that a plane algebraic curve of given degree can have? Lefschetz’s thesis, On the existence of loci with given singularities, earned his doctorate summa cum laude in 1911, after one year of graduate study. For the rest of his life he wrote and lectured with artificial hands.

Nebraska and Kansas, 1911–1924

Lefschetz took an assistantship at the University of Nebraska in Lincoln in 1911, and became an instructor there. He became an American citizen on 17 June 1912. On 3 July 1913 he married Alice Berg Hayes, whom he had met at Clark, where she was also a graduate student in mathematics. She helped him in daily life and in his work for the rest of his career. That year he moved to the University of Kansas at Lawrence, where he rose through the ranks to full professor in 1923.

Kansas was far from any research centre. There Lefschetz read the great works of the Italian and French schools, above all Émile Picard and Georges Simart’s Théorie des fonctions algébriques de deux variables indépendantes and Henri Poincaré’s papers on analysis situs, and set out to combine them. Picard had studied algebraic surfaces through integrals and their periods; Poincaré had created the topology needed to understand those periods. Lefschetz used Poincaré’s topology to prove theorems about algebraic surfaces and higher-dimensional varieties that the analysts and the Italian geometers could not reach.

In 1919 the Paris Academy of Sciences awarded him the Prix Bordin for a memoir connecting topology with Francesco Severi’s theory of the base of the group of divisors on a surface. An English version appeared in 1921 in the Transactions of the American Mathematical Society as On certain numerical invariants of algebraic varieties with application to abelian varieties; it won the Bôcher Memorial Prize. In 1924 he published the monograph L’analysis situs et la géométrie algébrique in Paris. The same year Princeton invited him as a visiting professor. Lessons 04 to 06 present the mathematics of this Kansas period.

Pencils and vanishing cycles

For the general reader. To understand a complicated shape, slice it. Lefschetz studied a complex algebraic variety by cutting it with a moving family of hyperplanes, a pencil. Almost every slice is smooth and has the same shape; finitely many slices are pinched at one point, where a small sphere in the slice shrinks to nothing. Keeping track of these vanishing spheres, and of how the slices twist as one moves around a pinch, tells you the topology of the whole variety.

Lefschetz pencils

Let \(X\subset\mathbb P^N\) be a smooth projective variety of complex dimension \(n\). A pencil of hyperplanes \(\{H_t\}_{t\in\mathbb P^1}\) is the family of hyperplanes containing a fixed codimension-2 linear space \(A\), its axis. Put \(X_t=X\cap H_t\) and \(B=X\cap A\).

Near an ordinary double point, the nearby smooth fibre \(z_1^2+\cdots+z_n^2=\varepsilon\) contains the real sphere \(\{z\in\sqrt\varepsilon\,\mathbb R^n:\sum z_i^2=\varepsilon\}\cong S^{n-1}\), which collapses to the singular point as \(\varepsilon\to0\). Its class \(\delta_i\in H_{n-1}(X_t)\) is the vanishing cycle. The nearby fibre retracts onto this sphere locally, so passing through a singular fibre changes the Euler characteristic by \(\chi(S^{n-1})-1=(-1)^{n-1}\).

Monodromy

Fix a regular value \(t_0\). Moving \(t\) once around a small loop enclosing \(t_i\) carries \(X_{t_0}\) back to itself and induces an automorphism \(T_i\) of \(H_{m}(X_{t_0};\mathbb Z)\), \(m=n-1\).

Picard had found the formula for surfaces, where the fibres are curves and \(T_i\) is a Dehn twist about the vanishing circle. Lefschetz proved it in all dimensions and used it to analyse \(H_*(X)\) by induction on dimension. Pierre Deligne’s 1974 proof of the Riemann hypothesis over finite fields rests on the same structure, transposed to \(\ell\)-adic cohomology.

Euler characteristics

Blowing up along \(B\) replaces \(B\) by \(B\times\mathbb P^1\), so \(\chi(\tilde X)=\chi(X)+\chi(B)\). Fibring \(\tilde X\) over \(\mathbb P^1\) with smooth fibre \(F\) and \(r\) nodal fibres gives \(\chi(\tilde X)=2\chi(F)+(-1)^n r\). Hence

The (1,1) theorem

For the general reader. On a surface, some loops and membranes can be drawn by algebraic curves, and some cannot. Which ones can? Lefschetz answered this for the two-dimensional classes in 1924: exactly those whose integrals satisfy a certain set of linear conditions. The generalization to higher dimensions is the Hodge conjecture, one of the Clay Millennium Problems.

Let \(X\) be a smooth complex projective variety. Every divisor \(D\), a formal sum of hypersurfaces, has a fundamental class \([D]\in H^2(X;\mathbb Z)\), equal to the first Chern class \(c_1(\mathcal O(D))\). The image of the group of divisors is the Néron–Severi group \(\mathrm{NS}(X)\), and its rank \(\rho(X)\) is the Picard number. By Hodge theory, \(H^2(X;\mathbb C)=H^{2,0}\oplus H^{1,1}\oplus H^{0,2}\).

Lefschetz stated and proved the theorem for surfaces in the language of integrals: a 2-cycle is algebraic exactly when the periods of all holomorphic 2-forms over it vanish. His proof used normal functions, integrals over the curves of a Lefschetz pencil. Hodge recast it in terms of harmonic forms, and in 1953 Kunihiko Kodaira and Donald Spencer gave the short proof now standard.

Proof via the exponential sequence

The exact sequence of sheaves \(0\to\mathbb Z\to\mathcal O_X\xrightarrow{\exp(2\pi i\,\cdot)}\mathcal O_X^\times\to0\) gives

The second map is the projection of \(H^2(X;\mathbb Z)\subset H^2(X;\mathbb C)\) onto the \((0,2)\) component. So \(\xi\) is \(c_1\) of a holomorphic line bundle exactly when its \((0,2)\) part is zero. For a real class this forces the \((2,0)\) part to vanish too, by complex conjugation, so \(\xi\in H^{1,1}\). On a projective variety every holomorphic line bundle is \(\mathcal O(D)\) for a divisor \(D\) (by Serre’s GAGA, or directly by Kodaira’s vanishing theorem), which completes the proof.

The Hodge conjecture

Hodge asked whether, more generally, every rational class in \(H^{2p}(X;\mathbb Q)\cap H^{p,p}\) is a rational combination of classes of subvarieties of codimension \(p\). The (1,1) theorem is the case \(p=1\). By the hard Lefschetz theorem (Lesson 06), it also gives the case \(p=n-1\), curves on an \(n\)-fold. The general case is open.

Hyperplane sections and hard Lefschetz

For the general reader. Slice a complex variety by a hyperplane. Lefschetz found that the slice carries almost all the topology of the whole variety in low dimensions, and that repeated slicing sets up a perfect symmetry between the low-dimensional and high-dimensional holes. These two theorems are now among the most used facts about projective varieties, and their analogues drive much of modern algebraic geometry and combinatorics.

Lefschetz’s argument in 1924 used a pencil and the vanishing cycles of Lesson 04. Complete proofs came in 1959 from Aldo Andreotti and Theodore Frankel and from Raoul Bott, using Morse theory: \(X\setminus Y\) is a smooth affine variety of dimension \(n\), and the squared distance from a generic point is a Morse function on it whose critical points all have index at most \(n\). So \(X\setminus Y\) has the homotopy type of a CW complex of dimension \(\le n\), and duality gives the theorem.

Defining the primitive classes \(P^{n-k}=\ker\big(L^{k+1}:H^{n-k}\to H^{n+k+2}\big)\), one gets the Lefschetz decomposition \(H^j=\bigoplus_{i\ge0}L^iP^{j-2i}\). In modern terms, \(L\) is part of an action of \(\mathfrak{sl}_2\) on \(H^*(X)\). Lefschetz’s topological proof had a gap; W. V. D. Hodge gave the first complete proof, via harmonic forms, and Deligne proved the analogue in \(\ell\)-adic cohomology over finite fields in 1980.

The fixed-point theorem

For the general reader. Stir a cup of coffee and let it settle: some point of the liquid ends where it started. L. E. J. Brouwer proved this for the disc. Lefschetz found a single number, computed from the holes of a space and the way a map moves them, that forces a fixed point whenever it is not zero, and that counts fixed points with signs when they are isolated.

Let \(X\) be a compact polyhedron (or a retract of one, such as a compact manifold) and \(f:X\to X\) continuous. The Lefschetz number of \(f\) is

It depends only on the homotopy class of \(f\), and \(\Lambda(\mathrm{id})=\chi(X)\).

Lefschetz proved the first version for compact orientable manifolds in 1923, extended it to manifolds with boundary in 1927 using relative homology, and to compact locally connected spaces in 1936. Heinz Hopf gave the version for polyhedra and the index formula in 1928–29.

Proof sketch

Suppose \(f\) has no fixed point. Then \(d(x,f(x))\ge\varepsilon>0\). Subdivide \(X\) so finely that every simplex has diameter less than \(\varepsilon/3\), and take a simplicial approximation \(g\) of \(f\) on a further subdivision. Then no simplex \(\sigma\) is carried into itself: \(g(\sigma)\cap\sigma=\varnothing\). So the chain map \(g_\#:C_k\to C_k\) has zero diagonal entries in the basis of simplices, and \(\operatorname{tr}g_\#=0\) in each degree. The Hopf trace formula says that for any chain map \(\phi\) of a finite complex,

which follows from the exact sequences \(0\to Z_k\to C_k\to B_{k-1}\to0\) and \(0\to B_k\to Z_k\to H_k\to0\) and additivity of traces. Hence \(\Lambda(f)=\Lambda(g)=0\).

Counting points over finite fields

For a variety over a finite field, the fixed points of the Frobenius map are the rational points. Weil saw that a Lefschetz formula in a suitable cohomology would give his conjectures (see the Weil course, Lesson 06). Grothendieck’s étale cohomology supplied that formula, \(\#X(\mathbb F_{q^m})=\sum_k(-1)^k\operatorname{tr}\big(F^m\mid H^k_c(X_{\bar{\mathbb F}_q};\mathbb Q_\ell)\big)\), the Grothendieck–Lefschetz trace formula.

Intersection, duality, coincidence

For the general reader. On a surface, two curves in general position cross at finitely many points, and the count with signs does not change when the curves move. Lefschetz turned this into an algebra for manifolds of any dimension: the intersection ring. He extended Poincaré’s duality to manifolds with boundary, and read fixed points as the places where the graph of a map crosses the diagonal.

In 1926 Lefschetz defined, for a closed oriented \(n\)-manifold \(M\), an intersection product \(H_p(M)\times H_q(M)\to H_{p+q-n}(M)\), by moving cycles into general position and intersecting them. This made \(H_*(M)\) a ring. Under Poincaré duality it corresponds to the cup product on cohomology, which was defined later (by Alexander, Čech and Whitney in the mid-1930s) and is now the usual way to present it.

Fixed points as intersections

For \(f:M\to M\) on a closed oriented manifold, let \(\Gamma_f=\{(x,f(x))\}\) and \(\Delta=\{(x,x)\}\) in \(M\times M\). Their intersection points are the fixed points of \(f\), and Lefschetz showed

More generally, for maps \(f,g:M\to N\) between closed oriented manifolds of the same dimension, he defined a coincidence number \(\Lambda(f,g)\), computed from \(f_*\) and \(g_*\) through Poincaré duality, which is non-zero only if \(f(x)=g(x)\) for some \(x\). The fixed-point theorem is the case \(N=M\), \(g=\mathrm{id}\).

Lefschetz numbers of iterates

For the general reader. Apply a map again and again, and ask how many points return to where they started after one step, two steps, a hundred steps. Lefschetz numbers of the iterates answer this with signs, and packaging them into one generating function gives a zeta function. The same construction, applied to the Frobenius map, is the zeta function of a variety over a finite field.

Let \(f:X\to X\) be a map of a compact polyhedron. The zeta function built from the Lefschetz numbers of its iterates, now called the Lefschetz zeta function, is

Proof. \(\Lambda(f^m)=\sum_k(-1)^k\operatorname{tr}(f_k^m)\), since \((f^m)_*=f_k^m\). For a matrix \(A\), \(\sum_{m\ge1}\operatorname{tr}(A^m)\,t^m/m=\sum_i\sum_m(\lambda_it)^m/m=-\sum_i\log(1-\lambda_it)=-\log\det(I-tA)\), where \(\lambda_i\) are the eigenvalues. Exponentiate and take the alternating product. \(\square\)

The formula has exactly the shape of the rationality statement in the Weil conjectures, with odd-degree factors in the numerator and even-degree factors in the denominator. That is no accident: when \(f\) is Frobenius and the fixed points are counted in étale cohomology, \(\zeta_f\) is the zeta function of the variety. In dynamics, Stephen Smale and others used \(\zeta_f\) from the 1960s to study the growth of periodic orbits.

Nonlinear differential equations

For the general reader. Many systems in engineering, such as electrical circuits, ship stabilizers and autopilots, are described by nonlinear equations that cannot be solved by formula. The useful questions are qualitative: does the system settle down, oscillate, or blow up? From 1943, Lefschetz turned to these questions. He wrote the first modern geometric textbook on them and trained the group that made stability theory and control a branch of American mathematics.

Consider \(\dot x=F(x)\) on \(\mathbb R^n\) with \(F\) of class \(C^1\) and \(F(0)=0\). The origin is stable if solutions starting near 0 stay near 0, and asymptotically stable if moreover they tend to 0.

Aleksandr Lyapunov’s direct method dates from 1892. Lefschetz’s collaborator Joseph LaSalle stated the invariance principle around 1960, and their book Stability by Liapunov’s Direct Method with Applications (1961) made the method standard in engineering. Lefschetz’s own Differential Equations: Geometric Theory (1957) presented the theory in the spirit of Poincaré: phase portraits, critical points, limit cycles, and the topology of the plane.

From 1950 to 1960 Lefschetz edited the five volumes of Contributions to the Theory of Nonlinear Oscillations in the Annals of Mathematics Studies. He also drew attention to the Soviet school of stability and control and to the notion of structural stability, a system whose phase portrait survives small perturbations, which became central in the work of Stephen Smale and Maurício Peixoto.

Princeton, 1924–1953

Lefschetz came to Princeton as a visiting professor in 1924, became associate professor in 1925 and full professor in 1927, and was Henry Burchard Fine Research Professor from 1933 until he retired in 1953. He chaired the department from 1945 to 1953. Princeton in these years, with Hermann Weyl, John von Neumann, James Alexander, Oswald Veblen and, from 1933, the Institute for Advanced Study in the same building, was the centre of topology and of much else in mathematics.

His books shaped the subject. Topology (1930) and Algebraic Topology (1942), both in the AMS Colloquium series, gave the field its name and much of its vocabulary. Introduction to Topology followed in 1949 and Algebraic Geometry in 1953. The last contains the Lefschetz principle, the informal rule that statements of algebraic geometry true over \(\mathbb C\) hold over every algebraically closed field of characteristic 0. He also helped bring Oscar Zariski from Italy to America in 1927, and talked mathematics with him for a decade.

From 1928 to 1958 he edited the Annals of Mathematics, and he made it one of the leading journals in the world. He was president of the American Mathematical Society in 1935–36. As a teacher he was forceful and demanding. His doctoral students at Princeton included Norman Steenrod, Albert W. Tucker, John Tukey, Ralph Fox, Paul A. Smith, Clifford Dowker, Henry Wallman, Shaun Wylie, Edward Begle, Richard Bellman, Felix Browder, John McCarthy and Ralph Gomory. Their later fields, from homotopy theory and knot theory to statistics, dynamic programming, artificial intelligence and integer programming, show how widely his influence ran. A saying among Princeton students held that he never stated a false theorem nor gave a correct proof: his intuition was sure even when his arguments were not.

In 1943 he became a consultant to the US Navy at the David Taylor Model Basin, where he worked with the engineer Nicholas Minorsky on the stabilization of ships. He was nearly sixty when he took up differential equations, the subject of his third career (Lesson 10).

Mexico, Baltimore, Providence; legacy

In 1944 Lefschetz began visiting the Institute of Mathematics of the National University of Mexico. From 1953 to 1966 he spent most winters in Mexico City, lecturing, organizing seminars and sending students to study abroad. His Mexican students and protégés included José Adem, Samuel Gitler, Emilio Lluis, Santiago López de Medrano and Alberto Verjovsky, and the school of topology he fostered remains strong. Mexico awarded him the Order of the Aztec Eagle in 1964.

After the launch of Sputnik in 1957, he came out of retirement to lead a mathematics centre for differential equations at the Research Institute for Advanced Studies in Baltimore, the research arm of the Glenn L. Martin Company. Joseph LaSalle and others joined him there. In 1964 the group moved to Brown University in Providence, where it became the Lefschetz Center for Dynamical Systems. He remained at Brown until 1970. He died in Princeton on 5 October 1972, aged 88.

He was elected to the National Academy of Sciences in 1925 and the American Philosophical Society in 1929, and became a Foreign Member of the Royal Society in 1961. He received the Feltrinelli Prize of the Accademia dei Lincei in 1956, the National Medal of Science in 1964, and in 1970 the first Leroy P. Steele Prize of the AMS. Hodge said that Lefschetz showed algebraic geometers that topology was indispensable to them.

Legacy

• Topology of algebraic varieties: pencils, vanishing cycles, the hyperplane and hard Lefschetz theorems, and the (1,1) theorem underlie Hodge theory, Deligne’s proof of the Weil conjectures, and intersection cohomology.
• Algebraic topology: the fixed-point and coincidence theorems, Lefschetz duality, the intersection ring, and the name of the subject.
• Dynamics and control: the American school of qualitative ODE theory and stability, continued at the Lefschetz Center at Brown.
• Hard Lefschetz far from geometry: analogues of hard Lefschetz proved by Richard Stanley (1980), and by June Huh and collaborators in the 2010s, resolved conjectures in combinatorics on polytopes and matroids.
• Names: Lefschetz number, Lefschetz pencil, Lefschetz fibration, Lefschetz duality, Lefschetz principle, Lefschetz zeta function, Picard–Lefschetz theory.

Connections

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

Teachers and advisors

• William Edward Story 1911

Students

• Albert W. Tucker
• Clifford Dowker
• Edward Begle
• Felix Browder
• Henry Wallman
• John McCarthy
• John Tukey
• José Adem
• Norman Steenrod
• Paul A. Smith
• Ralph Fox
• Ralph Gomory
• Richard Bellman
• Shaun Wylie

Influenced by

• Henri Poincaré
• Émile Picard
• Francesco Severi

Influenced

• André Weil
• Pierre Deligne
• W. V. D. Hodge

Colleagues and collaborators

• Heinz Hopf
• Hermann Weyl
• John von Neumann
• Oscar Zariski 1927-1937
• Pavel Alexandrov 1927-1928
• James Waddell Alexander
• Joseph LaSalle 1957-1970
• Nicholas Minorsky 1943
• Oswald Veblen

Also mentioned in this life

• L. E. J. Brouwer
• Leonhard Euler
• Pierre de Fermat
• Georg Frobenius
• Alexander Grothendieck
• Aleksandr Lyapunov
• Bernhard Riemann
• Jean-Pierre Serre

Fields

• algebraic geometry 17 lives
• algebraic topology 6 lives
• differential equations 13 lives

On CoursesHub

• Homology & Cohomology · Differential Geometry
• ODEs / PDEs / Green Functions · Physical Mathematics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Solomon Lefschetz, MacTutor History of Mathematics.
2. Solomon Lefschetz, Wikipedia.
3. P. Griffiths, D. Spencer and G. Whitehead, Solomon Lefschetz 1884–1972, Biographical Memoirs of the National Academy of Sciences 61 (1992).
4. S. Lefschetz, Reminiscences of a mathematical immigrant in the United States, Amer. Math. Monthly 77 (1970), 344–350.
5. Solomon Lefschetz: ensayo biográfico, Instituto de Matemáticas, UNAM.

Mathematics

1. S. Lefschetz, On certain numerical invariants of algebraic varieties with application to abelian varieties, Trans. Amer. Math. Soc. 22 (1921), 327–482.
2. S. Lefschetz, L’analysis situs et la géométrie algébrique, Gauthier-Villars, 1924.
3. S. Lefschetz, Intersections and transformations of complexes and manifolds, Trans. Amer. Math. Soc. 28 (1926), 1–49.
4. S. Lefschetz, Topology, AMS Colloquium Publications 12, 1930; Algebraic Topology, AMS Colloquium Publications 27, 1942.
5. S. Lefschetz, Differential Equations: Geometric Theory, Interscience, 1957.
6. J. LaSalle and S. Lefschetz, Stability by Liapunov’s Direct Method with Applications, Academic Press, 1961.
7. K. Lamotke, The topology of complex projective varieties after S. Lefschetz, Topology 20 (1981), 15–51.
8. V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps, vol. II, Birkhäuser, 1988.
9. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, 1978.
10. A. Hatcher, Algebraic Topology, Cambridge University Press, 2002 (Section 2.C on the fixed-point theorem).
11. Lefschetz fixed-point theorem, Lefschetz hyperplane theorem and Hard Lefschetz theorem, Wikipedia.