A Tribute to Robert Wayne Thomason 1952–1995
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K-Theory 12: 1–2, 1997. 1 c 1997 Kluwer Academic Publishers. Printed in the Netherlands. A Tribute to Robert Wayne Thomason 1952–1995 The current and following issue of -Theory are dedicated to the memory of one of its former editors Robert Thomason. Robert Wayne Thomason was born in Tulsa Oklahoma on 5 November 1952. While in high school, his talents and interest in mathematics were recognized. He attended Michigan State University during 1971–1973 and took 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 thesis, completed in 1977 under the guidance of 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 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 appoint- ment 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.
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Transcript of A Tribute to Robert Wayne Thomason 1952–1995
K-Theory 12: 1–2, 1997. 1c 1997 Kluwer Academic Publishers. Printed in the Netherlands.
A Tribute to Robert Wayne Thomason1952–1995
The current and following issue of K-Theory are dedicated to the memory of oneof its former editors Robert Thomason.
Robert Wayne Thomason was born in Tulsa Oklahoma on 5 November 1952.While in high school, his talents and interest in mathematics were recognized. Heattended Michigan State University during 1971–1973 and took advantage of itsflexible undergraduate honors mathematics program. While there, he published hisfirst paper, on point set topology. In 1973, he enrolled as a graduate student inthe mathematics department at Princeton University. There he became interested incategorical and simplicial methods in homotopy theory, an area of mathematics thatwould retain his interest for the rest of his life. His thesis, completed in 1977 underthe guidance of John Moore, shows that the geometric realization of a diagramof categories is homotopy equivalent to the geometric realization of the cofiberedcategory of the diagram, also called the lax colimit. This result has become astandard tool in the homotopy theory of categories.
In 1977, Thomason discovered his first major result, namely that all infinite loopspaces produce equivalent output. In order to straighten out the technical detailsof his insight, he enlisted the help of Peter May who recalls their cooperation asa ‘delightful interaction’. Their paper reduces the characterization of infinite loopspaces to just one axiom: the ‘group completion’ axiom.
The next two years were spent as a Moore Instructor at MIT. During thistime, Thomason studied the homotopy theory of categories, with emphasis onsymmetric monoidal categories. His results include a proof that the homotopytheory of small categories does not depend on geometric realization, because thecategory of all small categories is a closed model category. He also discoveredseveral homotopy colimit constructions for symmetric monoidal categories whichyield the corresponding homotopy colimits for spectra.
In 1979, Thomason went to the University of Chicago under a three year appoint-ment as a Dickson Assistant Professor. Soon after arriving, he began a four yeareffort 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 solutionof the Quillen–Lichtenbaum conjecture when the Bott element is inverted. Thisimportant result is the basis for much current research on the conjecture.
Thomason formulated the ‘homotopy limit problem’ as a generalization of hisapproach to the Quillen–Lichtenbaum conjecture, and several other conjectures aswell. This problem is dual to the homotopy colimit problem solved in his thesis.
corr. By john *INTERPRINT* J.V. Pip No.: 133053 MATHKAPkthe283.tex; 27/08/1997; 8:21; v.7; p.1
2 ANTHONY BAK AND CHARLES WEIBEL
When a group acts on a topological object, it asks how close the fixed point set isto the homotopy fixed point set. When the group is the Galois group of a field, onerecovers 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 Chica-go position in June 1980. For the next two years, he held an irregular appointmentat MIT and then was a member of IAS for a year.
In 1983, Thomason went to Johns Hopkins University. During his first threeyears there, he wrote a series of papers on equivariant algebraicK-theory. In 1985,he was awarded a Sloan Fellowship. He then began a program to solve the problemsleft over from Grothendieck’s SGA6, in particular the dependence of theK-theoryof a scheme on the derived category of vector bundles of the scheme and the effectof localization onK-theory. His solution of this problem in 1988 was his proudestachievement.
In recognition of his work above and on the Quillen–Lichtenbaum conjecture,Thomason was chosen to address the International Congress of Mathematicians inKyoto in 1990.
In the same year, he accepted an invitation from K-Theory to join its EditorialBoard. His participation on the board is characterized by his intense involvementin the referreeing process of papers under his supervision, a process which isexceptionally 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 finalsix years of his life. While in Paris, he helped run the monthly K-theory seminarand wrote six more articles. In the last year of his life, Thomason was able tosolve Grothendieck’s problem about lifting a homotopy structure from a categoryto functor categories. His main ideas centered around a modification of Quillen’saxioms for a closed model category. Unfortunately, he did not have time to writethis up.
Robert Thomason died in his Paris apartment in October, 1995 from diabeticshock.
ANTHONY BAK,CHARLES WEIBEL
kthe283.tex; 27/08/1997; 8:21; v.7; p.2
