Died: November 1995 in Paris, France

**Bob Thomason** entered Michigan State University in 1971 to read for a mathematics degree. He graduated in 1973 and then went to Princeton University to study for his doctorate. His supervisor at Princeton was John Moore and he wrote a dissertation on category theory in which he produced results which were to become fundamental tools in topology.

After graduating with his doctorate in June 1977 Thomason was appointed as Moore instructor at the Massachusetts Institute of Technology. This was a two year appointment and Thomason remained there until he took up a three year post as Dickson assistant professor at the University of Chicago in 1979.

Things did not go smoothly for Thomason, however, at the University of Chicago. While working on conjectures of Quillen-Lichtenbaum connecting K-theory to Étale cohomology Thomason produced what was first thought to be a remarkable proof. However, an error was found in 1980 and [4]:-

... Thomason began to feel uncomfortable about the scepticism expressed by others. Perceiving this as persecution, he resigned from his position at Chicago in June1980.

During the two years after Thomason resigned from Chicago he spent some time at the Massachusetts Institute of Technology and one year at the Institute for Advanced Study at Princeton. This was a mathematically profitable time for him and he was able to complete the results that had gone wrong while he was at Chicago along with some other major pieces of work.

Thomason's next post was at Johns Hopkins University where he was appointed in 1983. During the six years he spent there he produced a series of outstanding papers solving, among others, problems arising from Grothendieck's work in his paper with Berthelot and Illusie *Théorie des Intersections et Théorème de Riemann-Roch* (1971). For example in a 1983 paper he found a partial solution of Grothendieck's absolute cohomological purity conjecture.

During his time at Johns Hopkins University, Thomason was awarded a Sloan Fellowship which enabled him to spend the year 1987 at Rutgers University. Thomason spent three years working on the problems of Grothendieck referred to above. In [4] a most remarkable story is told regarding this work:-

While at Rutgers, he put everything in place except for one step ... On January22 1988, he had a dream in which his recently deceased friend Thomas Trobaugh told him how to solve the final step.... Awaking with a start, he worked out the argument for the missing step. In gratitude, he listed his friend as a coauthor of the resulting paper.

The importance of this work was recognised when Thomason was chosen to give an address at the International Congress of Mathematicians in Kyoto in 1990.

In October 1989 Thomason was appointed to a post in Max Karoubi's laboratory at the University of Paris VII. He continued to produce outstanding work, publishing six papers before his death in 1995. He had health problems which were to lead to his death [4]:-

Bob had diabetes and always had to strictly control what he ate. This made going to restaurants with Bob an awkward affair, because he would not eat something until he was sure it had no nutritional content. Early in November1995 [some accounts say late October], just before his43rd birthday, he went into diabetic shock and died in has apartment in Paris.

Thomason first important results concerned a proof that all infinite loop space machines produce equivalent output. Working with J P May he wrote a paper *The uniqueness of infinite loop space machines*. Quoting from the paper:-

An infinite loop space machine is a functor which constructs spectra out of ... space level data. There are many such machines known...;they are given by such widely disparate topological constructions that it is far from obvious that they turn out equivalent spectra when fed the same data. The purpose of this paper is to prove that all machines which satisfy certain reasonable properties do in fact turn out equivalent spectra.

Thomason then developed material which he had studied for his doctorate considering the homotopy theory of the category of small categories and the homotopy theory of the category of small symmetric monoidal categories.

We have already mentioned Thomason's results on the conjectures of Quillen-Lichtenbaum connecting K-theory to étale cohomology which he achieved during 1980-83. A reviewer of his paper *Algebraic K-theory and étale cohomology* (1985) wrote:-

This paper is one of the most important papers in algebraic K-theory since a paper by D Quillen ... in1972. Versions of the paper have circulated in manuscript form for several years and it is good to see it appear finally. The paper is important both for the results proved, and for the techniques used. The author pushes the applications of stable homotopy and homotopical algebra to algebraic K-theory and algebraic geometry further than anyone else and his methods have exerted considerable influence on other workers in the field.

Thomason's work during the next three years was on equivariant algebraic K-theory. He worked on the algebraic K-theory of algebraic group actions on schemes. He produced a theorem published in 1988 said to be:-

... one of the most important and powerful tools in algebraic K-theory. It is also one of the most sought-after. Much work has been done in attempts to extend Quillen's localization theorem to more general contexts, but none has even begun to approach the complete generality achieved in this paper.

A 1990 he contributed a paper (the one written after his dream with Thomas Trobaugh as coauthor) to *The Grothendieck Festschrift* which is described as a 'landmark paper'.

Thomason solved a problem in 1995 which had been posed by Grothendieck. The problem concerned lifting a homotopy structure. He lectured on this result in Genova only about three weeks before his death. He never wrote these results up for publication.

Weibel paints a picture of Thomason in [4]:-

Like many of his colleagues, Bob Thomason hated to waste energy on trivial matters, like fashion. He made the decision early in life to dress only in black clothing, thus simplifying that portion of his life. With his pointed goatee, he looked like a beat poet to outsiders, but mathematicians knew him as one of the greatest talents of his generation. Few have had the simultaneous grasp of topology, algebraic geometry and K-theory that Thomason did.

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

**September 1998**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Thomason.html]