Solomon Lefschetz (1884–1972) was a Russian-born American mathematician renowned for his foundational contributions to algebraic topology, algebraic geometry, and the theory of differential equations. Despite losing both hands in an industrial accident early in his career, he became one of the 20th century’s most influential mathematicians. He is best known for the Lefschetz fixed-point theorem, the Lefschetz hyperplane theorem, and his role in shaping the Princeton mathematics department. His work bridged topology and geometry, and he mentored numerous prominent mathematicians.
1 Early life and education
1.1 Childhood in Russia and emigration
Solomon Lefschetz was born on 3 September 1884 in Moscow, Russia, into a Jewish family. His father was a prosperous merchant, and the family moved to Paris when Lefschetz was a child. There he attended a French lycée, acquiring fluency in French. The family later relocated to the United States in 1902, settling in Philadelphia, where Lefschetz enrolled in an engineering program.
1.2 Engineering studies and accident
Lefschetz studied mechanical engineering at the University of Pennsylvania, graduating with a degree in 1905. He immediately began work as an engineer for the Westinghouse Electric Company in Pittsburgh. In 1907, a serious industrial accident occurred: while working on a high-voltage electrical circuit, he suffered severe burns that led to the amputation of both hands below the elbows. His career as a hands-on engineer was ended, but he retained full use of his arms and adapted by using prosthetic hooks.
1.3 Shift to mathematics at Clark University
After the accident, Lefschetz decided to pursue mathematics. He enrolled at Clark University in Worcester, Massachusetts, in 1908. The mathematics department there was small but active, and he was drawn to the work of William Edward Story. Lefschetz completed his master’s degree in 1909, then continued to doctoral studies.
1.4 Doctoral research and Ph.D.
For his Ph.D., Lefschetz investigated algebraic geometry and the theory of algebraic surfaces, a field deeply influenced by the Italian school. He wrote his dissertation under Story’s supervision, focusing on the existence of birational transformations. He received his doctorate in 1910. His early work already showed a geometric perspective that would later be crucial in topology.
2 Academic career
2.1 University of Nebraska and University of Kansas
Lefschetz spent the 1910–1911 academic year as an instructor at the University of Nebraska. He then moved to the University of Kansas, where he was an instructor and later an assistant professor until 1913. During this period, he began publishing papers on algebraic geometry and became interested in the emerging field of topology, then called analysis situs.
2.2 Princeton University (1913–1953)
2.2.1 Promotion to full professor
In 1913, Lefschetz was invited to Princeton University as a research associate. He quickly rose through the ranks, becoming an assistant professor in 1915, an associate professor in 1919, and a full professor in 1923. His promotion reflected his rapid production of major results, including his work on fixed-point theory and intersection numbers.
2.2.2 Chairmanship of the mathematics department
From 1924 to 1945, Lefschetz chaired the Princeton mathematics department. Under his leadership, the department became a leading center for topology and algebraic geometry. He recruited talented mathematicians such as John von Neumann, James Alexander, and later Albert Tucker. He also oversaw the creation of the Institute for Advanced Study’s mathematics faculty, even though the institute was administratively separate from Princeton University.
2.2.3 Mentorship of students and postdocs
Lefschetz was an energetic mentor, supervising more than thirty Ph.D. students. Among them were Robert Gunning, J. H. C. Whitehead, Norman Steenrod, and Richard Bellman. He also hosted many foreign visitors, especially from Europe and later from Latin America. His style was direct and often demanding, but he fostered a collaborative environment that produced lasting results.
2.3 Mexico years (1953–1966)
2.3.1 Work at the National Autonomous University of Mexico
After retiring from Princeton in 1953, Lefschetz accepted a position at the National Autonomous University of Mexico (UNAM) in Mexico City. He founded the Instituto de Matemáticas at UNAM and served as its director until 1966. There he continued his research, particularly on differential equations and the topological dynamics of nonlinear oscillations.
2.3.2 Influence on Latin American mathematics
Lefschetz’s presence in Mexico greatly strengthened mathematical research in Latin America. He trained a generation of Mexican mathematicians, including Roberto Vázquez, Samuel Gitler, and Guillermo Torres. He also organized conferences and exchanged ideas with mathematicians in Brazil and Argentina, helping to build an active regional community in topology and analysis.
3 Major mathematical contributions
3.1 Algebraic topology
3.1.1 Lefschetz fixed-point theorem
The Lefschetz fixed-point theorem (1926) is a foundational result that relates the number of fixed points of a continuous map on a compact polyhedron to a topological invariant, the Lefschetz number. It generalizes the Brouwer fixed-point theorem and provides a powerful tool for proving the existence of fixed points in many contexts. The theorem is central to modern topology and has applications in dynamical systems and functional analysis.
3.1.2 Lefschetz number and intersection theory
Lefschetz developed the concept of the Lefschetz number, a homological invariant that counts fixed points with multiplicities. This work was intimately connected to his earlier studies of intersection theory on algebraic varieties. He formulated intersection numbers for cycles on manifolds, laying groundwork for later developments by Andre Weil and others in the theory of algebraic cycles.
3.2 Algebraic geometry
3.2.1 Lefschetz hyperplane theorem
The Lefschetz hyperplane theorem (1924) is one of the most important results in the topology of algebraic varieties. It states that the homology groups of a projective algebraic variety can be inferred from those of a hyperplane section, at least in dimensions below the middle dimension. This result provides a fundamental link between the geometry of a variety and its topology, and it remains a cornerstone of algebraic geometry.
3.2.2 Picard–Lefschetz theory
Working on the monodromy of families of algebraic varieties, Lefschetz extended the work of Émile Picard on integrals of differential forms. The Picard–Lefschetz theory describes how the homology groups of a fiber change in a family with singular fibers, using the concept of vanishing cycles. This theory later influenced the development of Hodge theory and the study of singularities.
3.2.3 The Lefschetz principle
The Lefschetz principle is a metamathematical idea that a statement about algebraic varieties over an algebraically closed field of characteristic zero can be proved by considering the complex numbers, provided the statement is "algebraic" in nature. Lefschetz used this principle informally; later formalized, it became a tool for transferring results from complex geometry to other fields.
3.3 Ordinary differential equations
3.3.1 Stability theory and topological methods
Later in his career, Lefschetz turned to differential equations, particularly the study of stability and nonlinear oscillations. He applied topological methods such as fixed-point theorems to prove existence of periodic solutions. His book *Differential Equations: Geometric Theory* (1957) systematized these ideas.
3.3.2 Influence on dynamical systems
Lefschetz’s work, along with that of George Birkhoff and later Stephen Smale, helped establish the topological approach to dynamical systems. His fixed-point theorem remains a key tool for proving recurrence and the presence of attractors.
4 Key publications
4.1 Books
4.1.1 Topology (1930)
Lefschetz’s first major book, simply titled *Topology*, was published by the American Mathematical Society. It provided a comprehensive treatment of combinatorial topology and intersection theory, including his fixed-point theorem. The book helped standardize modern topology.
4.1.2 Algebraic Topology (1942)
This book (often called *Algebraic Topology* by Lefschetz) expanded on the earlier *Topology* and incorporated new developments such as homology theory and cohomology. It served as a standard reference for decades.
4.1.3 Contributions to the Theory of Nonlinear Oscillations (with others, 1950)
Co-authored with Nicholas Minorsky and others, this volume collected research on the theory of nonlinear oscillations. It marked Lefschetz’s engagement with differential equations and engineering applications.
4.2 Selected papers
Lefschetz published over 150 papers. Notable among them are “Continuous transformations of manifolds” (1923), which introduced the fixed-point theorem, and “On the existence of local coordinates” (with J. L. Walsh, 1930). His papers on Picard–Lefschetz theory appeared in the *Annals of Mathematics* in the 1920s.
5 Legacy and honors
5.1 Awards and memberships
5.1.1 National Academy of Sciences
Lefschetz was elected to the United States National Academy of Sciences in 1941. He was also a member of the American Academy of Arts and Sciences and a foreign member of several European academies.
5.1.2 American Mathematical Society presidency
He served as President of the American Mathematical Society from 1937 to 1939. His presidency saw the expansion of the Society’s publication program and its growing international influence.
5.2 Named concepts and terms
Concepts named after Lefschetz include:
- Lefschetz fixed-point theorem
- Lefschetz number
- Lefschetz hyperplane theorem
- Picard–Lefschetz theory
- Lefschetz principle
- Lefschetz surface (in algebraic geometry)
5.3 Influence on later mathematicians
Lefschetz directly influenced many students and colleagues. His intuitive geometric style was adopted by practitioners such as Raoul Bott, Michael Atiyah, and William Browder. In Mexico, his students established strong programs in topology and differential equations.
5.4 Posthumous recognitions
After his death, the Solomon Lefschetz Centennial Conference was held in 1984. The Solomon Lefschetz Award for Excellence in Teaching at the National Autonomous University of Mexico was named in his honor. His collected works were published in five volumes by Springer-Verlag.