**A TRIBUTE TO ROBERT WAYNE THOMASON (1952-1995)**

The current and following issue of K-Theory are dedicated to the
memory of one of its former editors Robert Thomason.
Robert Wayne Thomason was born in Tulsa Oklahoma on November 5,
1952. He attended Michigan State University during 1971-1973, taking
advantage of its flexible undergraduate honors mathematics
program. While there, he published his first paper, on
point set topology. In 1973 he enrolled as a graduate student in the
mathematics department at Princeton University. There he became interested
in categorical and simplicial methods in homotopy theory, an area of
mathematics that would retain his interest for the rest
of his life. His 1977 thesis, written under the guidance John Moore, shows
that the geometric realization of a diagram of categories
is homotopy equivalent to the geometric realization of the cofibered
category of the diagram, also called the lax colimit. This result has
become a standard tool in the homotopy theory of categories.
In 1977, Thomason discovered his first major result, namely that all
infinite loop spaces produce equivalent output. In order to straighten
out the technical details of his insight, he enlisted the help of
Peter May who recalls their cooperation as a "delightful
interaction". Their 1978 paper reduces the characterization of
infinite loop spaces to just one axiom: the ``group completion'' axiom.
The next two years were spent as a Moore Instructor at MIT. During
this time, Thomason studied the homotopy theory of categories, with
emphasis on symmetric monoidal categories. His results include a proof
that
the homotopy theory of small categories does not depend on geometric
realization, because the category of all small categories is a closed
model category. He also discovered several homotopy colimit
constructions for symmetric monoidal categories which yield the
corresponding homotopy colimits for spectra.
In 1979, Thomason went to the University of Chicago under a three year
appointment as a Dickson Assistant Professor. Soon after arriving, he
began a four year effort to solve the Quillen-Lichtenbaum
conjecture. Thomason's work in this area, alone and in joint work
with Dwyer, Snaith and Friedlander, ended in a solution of the
Quillen-Lichtenbaum conjecture when the Bott element is inverted. This
important result is the basis for much current research on the conjecture.
Thomason formulated the ``homotopy limit problem'' as a generalization
of
his approach to the Quillen-Lichtenbaum conjecture, and several other
conjectures as well. This problem is dual to the homotopy colimit
problem solved in his thesis. When a group acts on a topological
object, it asks how close the fixed point set is to the homotopy fixed
point set. When the group is the Galois group of a field, one recovers
the Quillen-Lichtenbaum conjecture.
After the collapse of an early attack on the Quillen-Lichtenbaum
conjecture, Thomason perceived the scepticism of others
as persecution and resigned his Chicago position in June 1980.
For the next two years, he held an irregular appointment at MIT and
then was a member of IAS for a year.
In 1983, Thomason went to John Hopkins University. During his first three
years there, he wrote a series of papers on equivariant algebraic K-theory.
In 1985, he was awarded a Sloan Fellowship. He then
began a program to solve the problems left over from Grothendiek's
SGA6, in particular the dependence of the K-theory of a
scheme on the derived category of vector bundles of the scheme and the
effect of localization on K-theory. His solution of this problem in
1988 was his proudest achievement.
In recognition of his work above and on the Quillen-Lichtenbaum
conjecture, Thomason was chosen to address the International Congress of
Mathematicians in Kyoto in 1990.
In the same year, he accepted an invitation from K-Theory to join its
Editorial Board. His participation on the board is characterized by
his intense involvement in the referreeing process of papers under his
supervision, a process which is exceptionally well documented in the
files left to this journal after his death.
In 1989, Thomason moved to the University of Paris, where he spent the
final six years of his life. While in Paris, Thomason helped run the
monthly K-theory seminar, and wrote six more articles. In the last year
of
his life, Thomason was able to solve Grothendieck's problem about
lifting a homotopy structure from a category to functor
categories. His main ideas centered around a modification of Quillen's
axioms for a closed model category. Unfortunately, he not have time to
write this up.
Robert Thomason died in his Paris apartment on October, 1995
from diabetic shock.
Anthony Bak, Charles Weibel