Department of Mathematics
The University of Chicago
Memorial from Volume 3, #1, of TopCom
Samuel Eilenberg, who made decisive contributions to topology and other areas of mathematics, died on Friday, Feb. 6 in New York City. He had been a leading member of the department of mathematics at Columbia University since 1947. His mathematical books, ideas and papers had a major influence.
Eilenberg was born in Poland in 1914. At the University of Warsaw he was a student of Borsuk in the active school of Polish Topology. His thesis, concerned with the topology of the plane, was published in Fundamenta Mathematica in 1936. Its results were well received both in Poland and in the USA. In 1938 he published there 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!
Early in 1939 Sammy's father told him "ammy, it doesn't look good here in Poland, Get out." He did, arriving in New York on April 23, 1939, and going at once to Princeton. At that University, Oswald Veblen and Solomon Lefschetz efficiently welcomed refugee mathematicians and found them suitable positions at American Universities. Sammy's work in topology was well known, so a position for him was found at the University of Michigan. There Ray Wilder had an active group of Topologists, including Norman Steenrod, then a recent Princeton Ph.D. Sammy immediately fitted in, did collaborative research (for example, with Wilder, O.G. Harrold and Dean Montgomery). His 1940 paper in the Annals formulated and codified the ideas of the "obstructions" recently introduced by Hassler Whitney. He also argued with Lefschetz. Finding the Lefschetz book obscure in its treatment of singular homology he provided an elegant and definitive treatment in the Annals (1949).
Sammy's idea was to dig deep and deeper till he got to the bottom of each issue. This I learned when I lectured at Ann Arbor about group extensions. I had calculated an example of the group of group extensions for an interesting factor group involving a prime number p. When I told Sammy this result he immediately saw that it answered a question of Steenrod about the regular cycles of the p-adic solenoid (Inside a solid Torus, wrap another one p times around, and so on, and infinitum). So Sammy and I stayed up all night to find out the reason for this unexpected appearance of group extensions. We found out more: it rested on a "universal coefficient theorem" which gave cohomology with any coefficient group G in terms of homology and an exact sequence involving Ext, the group of group extensions. Thus Sammy insisted on understanding this unexpected connection between algebra and topology. There was more there - the connection involved mapping topology into algebra, so we were forced to invent functors, natural transformations and categories to describe this. All told, this led to our fifteen joint papers.
They all involved the maxim: dig deeper and find out. For example, Hurewicz and Heinz Hopf had observed that the fundamental group of a space had effects on the higher homology and cohomology groups. Sammy, with his knowledge of his singular homology theory, had just the needed tools to understand this-- which resulted in our discovery of the cohomology of groups. Sammy saw that this idea went further, so he started Gerhard Hochschild on his study of the cohomology of algebra and then went on to write, with Henri Cartan, that very influential book on homological algebra-- the interest of many algebraists and provided the first book-presentation of French technique of spectral sequences.
Sammy applied his maxim in other connections. With Joe Zilber he developed the category of simplicial sets as a new type of space - using his singular simplices with face and degenerating operations. With Calvin Elgot he wrote about a recurring topic in logic. By himself he wrote two volumes on Automata Languages and Machines. And with Eldon Dyer he prepared two volumes (not yet published) on General and Categorical Topology.
Algebraic Topology was decisively influenced by Eilenberg's earlier 1952 work with Norman Steenrod on the Foundations of Algebraic Topology (Princeton University Press: now "books on demand"). At that time, there were many different and confusing versions of homology theory, some singular, some cellular. This book used categories to show that they all 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.
At Columbia University, Sammy took vigorous steps to build up the department. He trained many graduate students. For example, his students and postdocs in category theory include Applegate, Mike Barr, Jonathan Beck, David Buchsbaum, Alex Heller, Daniel Kan, Bill Lawvere, Fred Linton, Steve Schanuel, Myles Tierney and others. He was an inspiring teacher.
About two years ago, Sammy was felled by a stroke. It became hard for him to talk. In May 1997 I was able to visit him; he was lively and passed on to me a not-clearly understood proposal. He was then able to spend some time in his apartment on Riverside Drive. I think his message then to me was the same maxim: Keep on pressing those mathematical ideas. This is well illustrated by his life. His ideas: singular homology, categories, simplicial sets, generic acyclicity, obstructions and automata and the rest will live on.
Our fifteen joint research papers have been collected in the volume Eilenberg/MacLane, Collected Works, Academic Press Inc., New York 1988.