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Topology and K-Theory: Lectures by Daniel Quillen

Robert Penner
Publication Date: 
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[Reviewed by
Wojciech Dorabiala
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The book under review contains notes from a graduate student course on algebraic topology and K-theory taught by Daniel Quillen at the Massachusetts Institute of Technology during 1979-1980. Robert Penner does a fantastic job capturing the great mind of Quillen and preserving his work for future generations of mathematicians. The writing is true to Quillen's lecture style, reflecting his board-like presentation of formulas, diagrams, and proofs to give the reader the same feeling as when the lectures were presented forty years ago. The book consists of a set of 38 lectures that give an introduction to Quillen's fundamental work on higher algebraic K-theory. Across the lectures, readers with a basic background in abstract algebra are introduced to topics such as group cohomology, homological algebra, derived categories, simplicial homotopy theory and higher algebraic K-theory via Quillen's Q-construction. The topics covered in those lectures are available in modern textbooks; however, Quillen's informal style makes them more accessible to beginners. 
Quillen is without a doubt one of the most creative and influential mathematicians of his time. The book Topology and K-theory provides an excellent introduction to his work and the research for which he was awarded the Fields Medal in 1978, among others. It will be of great interest to graduate students and researchers interested in higher algebraic K-theory.
Wojciech Dorabiala is a professor of mathematics at Penn State – Altoona.  His areas of specialty are geometry topology and algebraic K-theory.