**Solomon Lefschetz** was a Russian born, Jewish mathematician who was the main source of the algebraic aspects of topology. His father Alexander Lefschetz and his mother Vera were both Turkish citizens but since Alexander Lefschetz worked as an importer he was required to travel a great deal. As a consequence the family decided to make a base for themselves in France where their children could be educated. Shortly after Solomon was born his family set up home in Paris.

Since Lefschetz was educated in France from a young age, French was his first language. He trained to be an engineer at the École Centrale in Paris from 1902 to 1905 and there attended lectures by Émile Picard and Paul Appell. However, not being a French citizen he would have found great difficulty obtaining an academic post in France. Fully understanding this, in November 1905 at the age of 21, Lefschetz went to the United States. For a few months he worked at the Baldwin Locomotive works, then from 1907 to 1910 he worked for Westinghouse Electric Company in Pittsburgh. He had the misfortune to lose both his hands in a laboratory accident in November 1907 when they were burnt off in a transformer explosion. He also lost his forearms and spent a while in hospital.

As one might imagine this accident had a major mental impact on Lefschetz in addition to the physical disability he suffered. The result was a deep depression, but the tragedy eventually pushed him towards mathematics, which was the right subject for him and, of course, it was fortunate for topology that he found his true love of mathematics. After teaching mathematics to apprentices at the Westinghouse Electric Company in 1910, he enrolled as a doctoral student in mathematics at Clark University in Worcester, Massachusetts, where he was a fellow from 1910 to 1911. While on the graduate program at Clark, Lefschetz met one of the mathematics students Alice Berg Hayes. They married on 3 July 1913, a year after he became an American citizen on 17 June 1912. Alice [1]:-

... helped him to overcome his handicap, encouraging him in his work and moderating his combative ebullience. They had no children.

Lefschetz received his Ph.D. in mathematics in 1911 with a thesis on algebraic geometry entitled *On the existence of loci with given singularities*.

That same year, 1911, Lefschetz was appointed an instructor in mathematics at the University of Nebraska in Lincoln, then, two years later, he was appointed to the University of Kansas in Lawrence. There he was promoted to assistant professor in 1916, to associate professor in 1919, and full professor in 1923. During these years he wrote a series of important papers on topology despite being out the mainstream of mathematical research. Lefschetz wrote of the importance of these years in his mathematical development:-

My years in the west with totally hermetic isolation played in my development the role of 'a job in a lighthouse' which Einstein would have every young scientist assume so that he may develop his own ideas in his own way.

His most important results from this period are contained in *On certain numerical invariants of algebraic varieties with application to abelian varieties*, which he had published in the *Transactions of the American Mathematical Society* in 1921, and in his famous monograph of 1924 *L'analysis situs et la géométrie algébrique*. Lefschetz explained with his famous quote:-

It was my lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry.

Hodge wrote (see [10] or [11]):-

Our greatest debt to Lefschetz lies in the fact that he showed us that a study of topology was essential for all algebraic geometers.

Poincaré had studied curves on a surface but Lefschetz pushed the ideas into much more general settings by building a theory of subvarieties of an algebraic variety. For his remarkable contributions during this period he was awarded the Prix Bordin by the Académie des Sciences in Paris in 1919 and the Bôcher Memorial Prize from the American Mathematical Society in 1923 for his 1921 paper we mentioned above.

In *Continuous transformations of manifolds* published in the *Proceedings of the U.S. National Academy of Sciences* in 1923 Lefschetz announced that he had:-

... new and far reaching methods...

for investigating continuous maps of manifolds and, in particular, their fixed points. He published his fixed point theorem for compact orientable manifolds in 1923, giving a fuller account of his famous fixed point theorem in *Intersections and transformations of complexes and manifolds* published in the *Transactions of the American Mathematical Society* in 1926. In this paper he promised more than the:-

... far reaching theory of the intersection of complexes on a manifold.

He wrote:-

With suitable restrictions the formulas derived are susceptible of extension to a wider range of manifolds, but this will be reserved for a later occasion.

In 1927 he fulfilled his promise extending his fixed point theorems to manifolds with boundary, and by this stage Brouwer's fixed point theorem became a special case. This paper is more than an extension, however, for in it he used matrices, in particular the trace of a matrix, to greatly simplify the formulae he had presented in his 1926 paper. He did further work on fixed point theorems studying the case of any finite complex in 1927 and any locally connected space in 1936.

On Alexander's recommendation, in 1924 Lefschetz went to Princeton as a visiting professor for a year. At the end of his visit in 1925 he was offered a permanent post at Princeton as associate professor which he happily accepted. He became Henry Fine Research Professor in 1933 filling the position left vacant when Veblen moved from Princeton University to the Institute for Advanced Study.

Sylvia Nasar writes [17]:-

When he first came to Princeton in the1920s, he often said he was an "invisible man". He was one of the first Jews on the faculty, loud, rude, and badly dressed to boot. People pretended not to see him in the hallways and gave him a wide berth at faculty parties. But Lefschetz had overcome far more formidable obstacles in his life than a bunch of prissy Wasp snobs.

Lefschetz worked on results which provided a deep generalisation of Émile Picard's theorems in function theory to several complex variables. Lefschetz was able to go further than Émile Picard and incorporate Poincaré's ideas. In doing this he developed a theory of algebraic topology of algebraic varieties of higher dimension. The word 'topology' comes from the title of a monograph written by Lefschetz in 1930. Another text which would have a huge influence on the development of the field was *Algebraic topology* which was published in 1942. In the course of his work he introduced many of what would be considered today the basic tools of algebraic topology [1]:-

He made extensive use of product spaces; he developed intersection theory, including the theory of the intersection ring of a manifold; and he made essential contributions to various kinds of homology theory, notably relative homology, singular homology, and cohomology.

Lefschetz had two artificial hands over which he always wore a shiny black glove. First thing every morning a graduate student had to push a piece of chalk into his hand and remove it at the end of the day. The students at Princeton made up a ditty about Lefschetz:-

Here's to Lefschetz, Solomon L.

Irrepressible as hell

When he's at last beneath the sod

He'll then begin to heckle God.

Saunders Mac Lane has added that:-

In my experience, Lefschetz was both obstreperous and enthusiastic - about research in mathematics.

The author of [2] writes:-

His enthusiasm and commanding personality enabled him to influence greatly the work of many of his juniors, so that they came to reverence him as the founder of their careers.

For Lefschetz, independent thinking and originality were what mattered in mathematical research. Unlike most mathematicians he had no respect for elegance and if something was to him clearly true, he would consider it at best a waste of time producing a rigorous argument to verify it. When a student proudly showed him a clever argument that he had produced to give a short proof of one of Lefschetz's theorems, rather than compliment the student, he is claimed to have retorted:-

Don't come to me with your pretty proofs. We don't bother with that baby stuff around here.

Even if there is little truth in a joke which circulated about Lefschetz, namely that he never wrote a correct proof or stated an incorrect theorem, there is an underlying truth in it reflecting on his style of mathematics.

Sylvia Nasar gives this vivid description of the impact Lefschetz had on Princeton [17]:-

Entrepreneurial and energetic, Lefschetz was the supercharged human locomotive that ... pulled the Princeton department out of genteel mediocrity right to the top. He recruited mathematicians with only one criterion in mind: research. His high-handed and idiosyncratic editorial policies made the Annals of Mathematics, Princeton's once-tired monthly, into the most revered mathematical journal in the world. He was sometimes accused of caving in to anti-Semitism for refusing to admit many Jewish students(his rationale being that nobody would hire them when they completed their degrees), but no one denies that he had brilliant snap judgement. He exhorted, bossed, and bullied, but with the aim of making the department great and turning his students into real mathematicians, tough like himself.

He was the editor of the *Annals of Mathematics* from 1928 to 1958, bringing it up to the standard of one of the very best world class journals. Steenrod wrote:-

The importance to American mathematicians of a first-class journal is that it sets high standards for them to aim at. In this somewhat indirect manner, Lefschetz profoundly affected the development of mathematics in the United States.

Lefschetz was a strong supporter of the American Mathematical Society serving as its President from 1935 to 1936.

During World War II, Lefschetz undertook applied mathematical work directed at the war effort. At the same time he heard of work being undertaken in the Soviet Union on applications of modern mathematical tools to applied mathematical problems. In 1944 these two influences inspired Lefschetz to return to his interest in engineering but now he had deep mathematical skills which he could bring to bear on the problems. He tackled problems related to dissipative nonlinear ordinary differential equations but did not take the usual approach of using linear theory to tackle nonlinear differential equations. Although initially his work was rather concrete, over the years it became more abstract as Lefschetz developed it further. In the end the ideas he developed became the foundation of a new branch of mathematics, namely global analysis.

Lefschetz had many students working in this area and, between 1950 and 1960, a series of important publications *Contributions to the theory of nonlinear oscillations* appeared in the *Annals of Mathematics Studies*, published by Princeton University Press. From this work by Lefschetz's school, came the two important concepts of structural stability and genericity. Lefschetz did remarkable work even well into his 80's. In his last book, he wrote on recent ideas outside his own area, namely the topology of Feynman integrals.

During the 1920s and 1930s Lefschetz was able to indulge his love of travel with many trips to European countries. In particular he loved to visit France, Italy and the Soviet Union. However the outbreak of World War II made European travel virtually impossible so Lefschetz had to find new places to visit. He chose Mexico, visiting the National University of Mexico for the first time in 1944. It was the first of many visits and he eventually fell into the habit of spending the summer months there every year. His contribution to mathematics in Mexico was, as a result of these visits, of major importance and he helped build a flourishing school there. His contributions were recognised when he was awarded the Order of the Aztec Eagle in 1964.

This was one of many honours which were awarded to Lefschetz. He received the Antonio Feltrinelli International Prize from the Accademia dei Lincei in 1956. He was awarded honorary degrees from the universities of Paris, Prague, Mexico, Clark, Brown, and Princeton. In 1964 he received the National Medal of Science:-

... for indomitable leadership in developing mathematics and training mathematicians, for fundamental publications in algebraic geometry and topology, and for stimulating needed research in nonlinear control processes.

**Article by:** *J J O'Connor* and *E F Robertson*

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