Topology and K-Theory

1. Front Matter

   Pages i-viii

   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

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.

• Topology
• K-Theory
• Algebraic Topology
• Category Theory
• Homological Algebra
• Quillen K-Theory

“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)

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

  Robert Penner

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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