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Topology and K-Theory

Lectures by Daniel Quillen

  • Book
  • © 2020

Overview

Authors:
  1. Robert Penner

    1. Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France

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  • Useful text for second-semester graduate student learning Algebraic Topology
  • Useful for students or researchers at any level as a preface to Quillen's three landmark papers
  • As a historical or sociological point, these informal notes provide a glimpse of Quillen's magnificent mind

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2262)

Part of the book sub series: History of Mathematics Subseries (HISTORYMS)

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Table of contents (38 chapters)

  1. Front Matter

    Pages i-viii
    Download chapter PDF

  2. Group Extensions and Cohomology

    • Robert Penner
    Pages 1-4

  3. Categories and Their Nerves

    • Robert Penner
    Pages 5-8

  4. Simplicial Objects

    • Robert Penner
    Pages 9-13

  5. Normalization and Conical Contractibility

    • Robert Penner
    Pages 15-19

  6. Effaceable \(\delta \)-Functors

    • Robert Penner
    Pages 21-24

  7. (Co)homology of Cyclic Groups

    • Robert Penner
    Pages 25-31

  8. An Application to the Schur–Zassenhaus Theorem

    • Robert Penner
    Pages 33-37

  9. The Yoneda Lemma

    • Robert Penner
    Pages 39-44

  10. Kan Formulae

    • Robert Penner
    Pages 45-49

  11. Abelian and Additive Categories

    • Robert Penner
    Pages 51-55

  12. Diagram Chasing in Abelian Categories

    • Robert Penner
    Pages 57-60

  13. Fibered and Cofibered Categories

    • Robert Penner
    Pages 61-65

  14. Examples of Fibered Categories

    • Robert Penner
    Pages 67-70

  15. Projective Resolutions

    • Robert Penner
    Pages 71-74

  16. Analogues of Homotopy Liftings

    • Robert Penner
    Pages 75-82

  17. The Mapping Cylinder and Mapping Cone

    • Robert Penner
    Pages 83-88

  18. Derived Categories

    • Robert Penner
    Pages 89-93

  19. The First Homotopy Property

    • Robert Penner
    Pages 95-100

  20. Group Completions and Grothendieck Groups

    • Robert Penner
    Pages 101-103
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Keywords

About this book

These are notes from a graduate student course on algebraic topology and K-theory given by Daniel Quillen at the Massachusetts Institute of Technology during 1979-1980. He had just received the Fields Medal for his work on these topics among others and was funny and playful with a confident humility from the start. These are not meant to be polished lecture notes, rather, things are presented as did Quillen reflected in the hand-written notes, resisting any temptation to change or add notation, details or elaborations. Indeed, the text is faithful to Quillen's own exposition, even respecting the {\sl board-like presentation} of formulae, diagrams and proofs, omitting numbering theorems in favor of names and so on. This is meant to be Quillen on Quillen as it happened forty years ago, an informal text for a second-semester graduate student on topology, category theory and K-theory, a potential preface to studying Quillen's own landmark papers and an informal glimpse of his great mind. The intellectual pace of the lectures, namely fast and lively, is Quillen himself, and part of the point here is to capture some of this intimacy. To be sure, much has happened since then from this categorical perspective started by Grothendieck, and Misha Kapranov has contributed an Afterword in order to make it more useful to current students.





Reviews

“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, MAA Reviews, February 19, 2023)

Authors and Affiliations

  • Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France

    Robert Penner

About the author

Robert Penner holds the Rene Thom Chair at the Institut des Hautes Etudes Scientifiques in Paris. His research extends across topology, geometry and combinatorics together with their applications to high energy physics and theoretical biology.  Among his previous books are
Combinatorics of Train Tracks (Princeton University Press), Discrete Mathematics (World Scientific) and Decorated Teichmueller Theory
(European Mathematical Society).


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