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More Concise Algebraic Topology More Concise Algebraic Topology −1 0 +1 “530-46909_Ch00_5P.tex” — 8/19/2011 — 11:24 — page i chicago lectures in mathematics series Editors: Spencer J. Bloch, Peter Constantin, Benson Farb, Norman R. Lebovitz, Carlos Kenig, and J. P. May Other Chicago Lectures in Mathematics titles available from the University of Chicago Press Simplical Objects in Algebraic Topology, by J. Peter May (1967, 1993) Fields and Rings, Second Edition, by Irving Kaplansky (1969, 1972) Lie Algebras and Locally Compact Groups, by Irving Kaplansky (1971) Several Complex Variables, by Raghavan Narasimhan (1971) Torsion-Free Modules, by Eben Matlis (1973) Stable Homotopy and Generalised Homology, by J. F. Adams (1974) Rings with Involution, by I.N. Herstein (1976) Theory of Unitary Group Representation, by George V. Mackey (1976) Commutative Semigroup Rings, by Robert Gilmer (1984) Infinite-Dimensional Optimization and Convexity, by Ivar Ekeland and Thomas Turnbull (1983) Navier-Stokes Equations, by Peter Constantin and Ciprian Foias (1988) Essential Results of Functional Analysis, by Robert J. Zimmer (1990) Fuchsian Groups, by Svetlana Katok (1992) Unstable Modules over the Steenrod Algebra and Sullivan’s Fixed Point Set Conjecture, by Lionel Schwartz (1994) Topological Classification of Stratified Spaces, by Shmuel Weinberger (1994) Lectures on Exceptional Lie Groups, by J. F. Adams (1996) Geometry of Nonpositively Curved Manifolds, by Patrick B. Eberlein (1996) Dimension Theory in Dynamical Systems: Contemporary Views and Applications, by Yakov B. Pesin (1997) Harmonic Analysis and Partial Differential Equations: Essays in Honor of Alberto Calderon´ , edited by Michael Christ, Carlos Kenig, and Cora Sadosky (1999) A Concise Course in Algebraic Topology, by J. P. May (1999) Topics in Geometric Group Theory, by Pierre de la Harpe (2000) Exterior Differential Systems and Euler-Lagrange Partial Differential Equations, by Robert Bryant, Phillip Griffiths, and Daniel Grossman (2003) Ratner’s Theorems on Unipotent Flows, by Dave Witte Morris (2005) Geometry, Rigidity, and Group Actions, edited by Benson Farb and David Fisher (2011) Groups of Circle Diffeomorphisms, by Andrés Navas (2011) −1 0 +1 “530-46909_Ch00_5P.tex” — 8/19/2011 — 11:24 — page ii More Concise Algebraic Topology localization, completion, and model categories J. P. May and K. Ponto the university of chicago press • chicago and london −1 0 +1 “530-46909_Ch00_5P.tex” — 8/19/2011 — 11:24 — page iii J. P. May is professor of mathematics at the University of Chicago; he is author or coauthor of many papers and books, including Simplicial Objects in Algebraic Topology and A Concise Course in Algebraic Topology in this series. K. Ponto is assistant professor of mathematics at the University of Kentucky; she is an algebraic topologist whose research has focused on topological fixed point theory. The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London © 2012 by The University of Chicago All rights reserved. Published 2012. Printed in the United States of America 201918171615141312 12345 ISBN-13: 978-0-226-51178-8 (cloth) ISBN-10: 0-226-51178-2 (cloth) Library of Congress Cataloging-in-Publication Data May, J. P. More concise algebraic topology : localization, completion, and model categories / J. P. May and K. Ponto. p. cm. — (Chicago lectures in mathematics series) ISBN-13: 978-0-226-51178-8 (hardcover : alk. paper) ISBN-10: 0-226-51178-2 (hardcover : alk. paper) 1. Algebraic topology. I. Ponto, Kate. II. Title. III. Series: Chicago lectures in mathematics. QA612.M388 2012 514'.2—dc22 2011012400 ∞ This paper meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). −1 0 +1 “530-46909_Ch00_5P.tex” — 8/19/2011 — 11:24 — page iv Contents Introduction xi Some conventions and notations xxi Acknowledgments xxvii PART 1 || Preliminaries: Basic Homotopy Theory and Nilpotent Spaces 1. Cofibrations and fibrations 3 1.1 Relations between cofibrations and fibrations 3 1.2 The fill-in and Verdier lemmas 7 1.3 Based and free cofibrations and fibrations 11 1.4 Actions of fundamental groups on homotopy classes of maps 15 1.5 Actions of fundamental groups in fibration sequences 18 1 2. Homotopy colimits and homotopy limits; lim 24 2.1 Some basic homotopy colimits 24 2.2 Some basic homotopy limits 31 2.3 Algebraic properties of lim1 35 2.4 An example of nonvanishing lim1 terms 38 2.5 The homology of colimits and limits 40 2.6 A profinite universal coefficient theorem 43 3. Nilpotent spaces and Postnikov towers 46 3.1 A -nilpotent groups and spaces 46 3.2 Nilpotent spaces and Postnikov towers 49 3.3 Cocellular spaces and the dual Whitehead theorem 50 3.4 Fibrations with fiber an Eilenberg-Mac Lane space 56 3.5 Postnikov A -towers 61 −1 0 +1 “530-46909_Ch00_5P.tex” — 8/19/2011 — 11:24 — page v vi / contents 4. Detecting nilpotent groups and spaces 69 4.1 Nilpotent actions and cohomology 69 4.2 Universal covers of nilpotent spaces 72 4.3 A -maps of A -nilpotent groups and spaces 74 4.4 Nilpotency and fibrations 76 4.5 Nilpotent spaces and finite type conditions 79 PART 2 || Localizations of Spaces at Sets of Primes 5. Localizations of nilpotent groups and spaces 87 5.1 Localizations of abelian groups 87 5.2 The definition of localizations of spaces 91 5.3 Localizations of nilpotent spaces 94 5.4 Localizations of nilpotent groups 97 5.5 Algebraic properties of localizations of nilpotent groups 101 5.6 Finitely generated T-local groups 106 6. Characterizations and properties of localizations 111 6.1 Characterizations of localizations of nilpotent spaces 111 6.2 Localizations of limits and fiber sequences 116 6.3 Localizations of function spaces 119 6.4 Localizations of colimits and cofiber sequences 121 6.5 A cellular construction of localizations 123 6.6 Localizations of H-spaces and co-H-spaces 125 6.7 Rationalization and the finiteness of homotopy groups 128 6.8 The vanishing of rational phantom maps 130 7. Fracture theorems for localization: groups 132 7.1 Global to local pullback diagrams 133 7.2 Global to local: abelian and nilpotent groups 137 7.3 Local to global pullback diagrams 140 7.4 Local to global: abelian and nilpotent groups 143 7.5 The genus of abelian and nilpotent groups 145 7.6 Exact sequences of groups and pullbacks 150 8. Fracture theorems for localization: spaces 154 8.1 Statements of the main fracture theorems 154 8.2 Fracture theorems for maps into nilpotent spaces 157 8.3 Global to local fracture theorems: spaces 161 −1 0 +1 “530-46909_Ch00_5P.tex” — 8/19/2011 — 11:24 — page vi contents / vii 8.4 Local to global fracture theorems: spaces 164 8.5 The genus of nilpotent spaces 165 8.6 Alternative proofs of the fracture theorems 170 9. Rational H-spaces and fracture theorems 175 9.1 The structure of rational H-spaces 175 9.2 The Samelson product and H∗(X; Q) 178 9.3 The Whitehead product 182 9.4 Fracture theorems for H-spaces 184 PART 3 || Completions of Spaces at Sets of Primes 10. Completions of nilpotent groups and spaces 191 10.1 Completions of abelian groups 191 10.2 The definition of completions of spaces at T 199 10.3 Completions of nilpotent spaces 202 10.4 Completions of nilpotent groups 207 11. Characterizations and properties of completions 214 11.1 Characterizations of completions of nilpotent spaces 214 11.2 Completions of limits and fiber sequences 217 11.3 Completions of function spaces 220 11.4 Completions of colimits and cofiber sequences 221 11.5 Completions of H-spaces 222 11.6 The vanishing of p-adic phantom maps 223 12. Fracture theorems for completion: groups 226 12.1 Preliminaries on pullbacks and isomorphisms 227 12.2 Global to local: abelian and nilpotent groups 229 12.3 Local to global: abelian and nilpotent groups 234 12.4 Formal completions and the adèlic genus 236 13. Fracture theorems for completion: spaces 243 13.1 Statements of the main fracture theorems 244 13.2 Global to local fracture theorems: spaces 246 13.3 Local to global fracture theorems: spaces 248 13.4 The tensor product of a space and a ring 254 13.5 Sullivan’s formal completion 257 13.6 Formal completions and the adèlic genus 260 −1 0 +1 “530-46909_Ch00_5P.tex” — 8/19/2011 — 11:24 — page vii viii / contents PART 4 || An Introduction to Model Category Theory 14. An introduction to model category theory 267 14.1 Preliminary definitions and weak factorization systems 268 14.2 The definition and first properties of model categories 274 14.3 The notion of homotopy in a model category 280 14.4 The homotopy category of a model category 286 15. Cofibrantly generated and proper model categories 292 15.1 The small object argument for the construction of WFSs 293 15.2 Compactly and cofibrantly generated model categories 299 15.3 Over and under model structures 301 15.4 Left and right proper model categories 305 15.5 Left properness, lifting properties, and the sets [X, Y] 309 16. Categorical perspectives on model categories 314 16.1 Derived functors and derived natural transformations 314 16.2 Quillen adjunctions and Quillen equivalences 319 16.3 Symmetric monoidal categories and enriched categories 323 16.4 Symmetric monoidal and enriched model categories 329 16.5 A glimpse at higher categorical structures 335 17. Model structures on the category of spaces 339 17.1 The Hurewicz or h-model structure on spaces 340 17.2 The Quillen or q-model structure on spaces 344 17.3 Mixed model structures in general 348 17.4 The mixed model structure on spaces 356 17.5 The model structure on simplicial sets 359 17.6 The proof of the model axioms 365 18.
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