**Samuel Eilenberg**'s father was educated at a Jewish school but became a brewer as he married into a family of brewers. Sammy, as Eilenberg was always called, studied at the University of Warsaw. It is not surprising that Eilenberg's interests quickly turned towards point set topology which, of course, was an area which flourished at the University of Warsaw at that time.

A remarkable collection of mathematicians were on the staff at the University of Warsaw while Eilenberg studied there. For example Mazurkiewicz, Kuratowski, Sierpinski, Saks and Borsuk taught there. Eilenberg was awarded his MA from the University of Warsaw in 1934. Then in 1936 he received his doctorate after studying under Borsuk. Mac Lane writes in [1]:-

His thesis, concerned with the topology of the plane, was published in Fundamenta Mathematicae in1936. Its results were well received both in Poland and the USA.

The second mathematical centre in Poland at this time was Lvov. It was there that Eilenberg met Banach, who led the Lvov mathematicians. He joined the community of mathematicians working and drinking in the Scottish Café and he contributed problems to the Scottish Book, the famous book in which the mathematicians working in the Café entered unsolved problems.

You can see a picture of the Scottish Café.

Most of Eilenberg's publications from this period were on point-set topology but there were signs, even at this early stage of his career, that he was moving towards more algebraic topics. Mac Lane writes [1]:-

In1938he published in[Fundamenta Mathematicae]another influential paper on the action of the fundamental group on the higher homotopy groups of a space. Algebra was not foreign to his topology!

This paper was an early sign that Eilenberg was moving into the area for which he has become famous. It was one of a truly remarkable collection of papers published by Eilenberg for, from his days as an undergraduate up until 1939 when he left Poland for the United States, he published 37 papers.

In 1939 Eilenberg's father convinced him that the right course of action was to emigrate to the United States. Once there he went to Princeton where Veblen and Lefschetz helped him to find a university post. This was not too long in coming and, in 1940, he was appointed as an instructor at the University of Michigan. This was an excellent place for Eilenberg to begin his teaching career in the United States for there he could interact with leading topologists. Wilder was on the staff at Ann Arbor and Steenrod, who had studied there earlier, continued to have close links and returned onto the staff at Ann Arbor in 1942.

In 1940 there was an important topology conference organised at Michigan. World War II was by this time dominating the international scene so the number of participants at the conference from outside the United States was much less than one would have otherwise expected. Eilenberg lectured at the conference on *Extension and classification of continuous mappings*.

Eilenberg was only an instructor for one year, then in 1941 he was promoted at assistant professor at the University of Michigan. In 1945 he was promoted again, this time to associate professor. He spent the year 1945-46 as a visiting lecturer at Princeton before being appointed as a full professor at the University of Indiana in 1946. After one year he moved to Columbia University in New York where he remained for the rest of his career. In 1948, the year after he took up the post at Columbia, Eilenberg became a US citizen. He married Natasa Chterenzon in 1960.

Perhaps the most obvious feature of Eilenberg's work was the amount which was done in collaboration with other mathematicians. One major collaboration was his work with Bourbaki. In 1949 André Weil was working at the University of Chicago and he contacted Eilenberg to ask him to collaborate on writing about homotopy groups and fibre spaces as part of the Bourbaki project. Eilenberg became a member of the Bourbaki team spending 1950-51 as a visiting professor in Paris and participating in the two week summer meetings until 1966. He had been awarded Fulbright and Guggenheim scholarships to fund his year in Paris.

One of the first collaborations which Eilenberg entered was with Mac Lane. The two first met in 1940 in Ann Arbor and from that time until about 1954 the pair produced fifteen papers on a whole range of topics including category theory, cohomology of groups, the relation between homology and homotopy, Eilenberg-Mac Lane spaces, and generic cycles. In 1942 they published a paper in which they introduced *Hom* and *Ext* for the first time. They introduced the terms functor and natural isomorphism and, in 1945, added the terms *category* and *natural transformation*.

Ann Arbor again provided the means to bring Eilenberg and Steenrod together. In 1945 they set out the axioms for homology and cohomology theory but they did not give proofs in their paper, leaving these to appear in their famous text *Foundations of algebraic topology* in 1952. Mac Lane writes in [1]:-

At that time there were many different and confusing versions of homology theory, some singular some cellular. The book used categories to show that they all could be described conceptually as presenting homology functors from the category of pairs of spaces to groups or to rings, satisfying suitable axioms such as "excision". Thanks to Sammy's insight and his enthusiasm, this text drastically changed the teaching of topology.

In fact Eilenberg had written a definitive treatment of singular homology and cohomology in a paper in the *Annals of Mathematics* in 1944. He had written this paper since he found the treatment of the topic by Lefschetz in his 1942 book unsatisfactory. In 1948 Eilenberg, in a joint paper with Chevalley, gave an algebraic approach to the cohomology of Lie groups, using the Lie algebra as a basic object. They showed that in characteristic zero the cohomology of a compact Lie group is isomorphic as an algebra to the cohomology of the corresponding Lie algebra.

Another collaboration of major importance was between Eilenberg and Henri Cartan. The two first met in 1947 and began to exchange ideas by letter in the following years. However as we mentioned above Eilenberg spent 1950-51 in Paris and it was during this time that they made remarkable progress. Henri Cartan writes in [1]:-

We went from discovery to discovery, Sammy having an extraordinary gift for formulating at each moment the conclusions that would emerge from the discussion. And it was always he who wrote everything up as we went along in precise and concise English. ... Of course, this work together took several years. Sammy made several trips to my country houses(in Die and in Dolomieu). Outside of our work hours he participated in our family life.

The outcome of this collaboration was the book *Homological algebra* the title being a term which the two mathematicians invented. Although they had completed the manuscript by 1953 it was not published until 1956. Hochschild reviewing the book wrote:-

The title "Homological Algebra" is intended to designate a part of pure algebra which is the result of making algebraic homology theory independent of its original habitat in topology and building it up to a general theory of modules over associative rings. ... The conceptual flavour of homological algebra derives less specifically from topology than from the general "naturalistic" trend of mathematics as a whole to supplement the study of the anatomy of any mathematical entity with an analysis of its behaviour under the maps belonging to the larger mathematical system with which it is associated. In particular, homological algebra is concerned not so much with the intrinsic structure of modules but primarily with the pattern of compositions of homomorphisms between modules and their interplay with the various constructions by which new modules may be obtained from given ones.

He also notes:-

The appearance of this book must mean that the experimental phase of homological algebra is now surpassed. The diverse original homological constructions in various algebraic systems which were frequently of an ad hoc and artificial nature have been absorbed in a general theory whose significance goes far beyond its sources. The basic principles of homological algebra, and in particular the full functorial control over the manipulation of tensor products and modules of operator homomorphisms, will undoubtedly become standard algebraic technique already on the elementary level.

We should mention another major two volume text which Eilenberg published in 1974 and 1976. This text was *Automata, languages, and machines* which was described by a reviewer as:-

... one of the most important events in the mathematical study of the foundations of computer science and in applied mathematics. The work includes a unifying mathematical presentation of almost all major topics of automata and formal language theory.

It was a topic which Eilenberg had been interested in from 1966 onwards and it is worth noting that it is one of the few major works by Eilenberg which he worked on alone. The book examines rational structures, that is those that can be recognised by a finite state automaton.

So far we have only talked about Sammy the mathematician. There was another side to Eilenberg however, for he was a dealer in the art world in which he was known as "Professor". He dealt in Indian art and he was a leading expert on the subject. Hyman Bass writes in [1]:-

Over the years Sammy gathered one of the world's most important collections of Southeast Asian art. His fame among certain art collectors overshadows his mathematical reputation. In a gesture characteristically marked by its generosity and elegance, Sammy in1987donated much of his collection to the Metropolitan Museum of art in New York, which in turn was thus motivated to contribute substantially to the endowment of the Eilenberg Visiting Professorship in Mathematics at Columbia university.

Eilenberg received many honours for his work. In particular we should mention the Wolf Prize which he shared with Selberg in 1986 and his election to the National Academy of Sciences.

Finally let us give a quote regarding Eilenberg's personality. Bass writes in [1]:-

Though his mathematical ideas may seem to have a kind of crystalline austerity, Sammy was a warm, robust, and very animated human being. For him mathematics was a social activity, whence his many collaborations. He liked to do mathematics on his feet, often prancing while he explained his thoughts. When something connected, one could read it in his impish smile and the sparkle in his eyes.

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

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