Topology and K-Theory - Lectures by Daniel Quillen

Description

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. 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). Preface......Page 6 Contents......Page 8 1 Group Extensions and Cohomology......Page 10 2 Categories and Their Nerves......Page 14 3 Simplicial Objects......Page 18 4 Normalization and Conical Contractibility......Page 23 5 Effaceable δ-Functors......Page 28 6 (Co)homology of Cyclic Groups......Page 32 7 An Application to the Schur–Zassenhaus Theorem......Page 39 8 The Yoneda Lemma......Page 44 9 Kan Formulae......Page 50 10 Abelian and Additive Categories......Page 55 11 Diagram Chasing in Abelian Categories......Page 60 12 Fibered and Cofibered Categories......Page 64 13 Examples of Fibered Categories......Page 69 14 Projective Resolutions......Page 73 15 Analogues of Homotopy Liftings......Page 77 16 The Mapping Cylinder and Mapping Cone......Page 85 17 Derived Categories......Page 91 18 The First Homotopy Property......Page 96 19 Group Completions and Grothendieck Groups......Page 102 20 Devissage and Resolution Theorems......Page 105 21 Exact Sequences of Homotopy Classes......Page 109 22 Spectral Sequences......Page 113 23 Spectral Sequences Continued......Page 119 24 Hyper-Homology Spectral Sequences......Page 125 25 Generalized Kan Formulae......Page 131 26 The Hochschild–Serre Spectral Sequence......Page 136 27 Resolution for Exact Categories......Page 141 28 K0 A .5-.5.5-.5.5-.5.5-.5 K0 A[T]......Page 147 29 Classifying Spaces......Page 152 30 Higher K-Groups......Page 155 31 The Category QmathcalM......Page 160 32 Homotopy Equivalence......Page 164 33 A Filtration of Q(mathcalPA)......Page 169 34 Bi-simplicial Sets and Dold–Thom......Page 172 35 Homology of Q(mathcalPA) and the Tits Complex......Page 177 36 Long Exact Sequences of K-Groups......Page 181 37 Localization......Page 186 38 The Plus Construction, K1 and K2......Page 191 Afterword by Mikhail Karpranov......Page 196 References......Page 198 Index......Page 199