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Lecture Notes in Mathematics a Collection of Informal Reports and Seminars Edited by A Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z(Jrich 304 A. K. Bousfield University of Illinois, Chicago, IL/USA D. M. Kan Massachusetts Institute of Technology, Cambridge, MA/USA Homotopy Limits, Completions and Localizations Springer-Verlag Berlin. Heidelberg • New York 1972 AMS Subject Classifications (1970): 18A30, 18A35, 55-02, 55D05, 55D10, 55D99 ISBN 3-540-06105-3 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-06105-3 Springer-Verlag New York • Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin - Heidelberg 1972. Library of Congress Catalog Card Number 72-95678. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr. Contents Part I. Completions and localizations §0. Introduction to Part I ....................... Chapter I. The R-completion of a space §l. Introduction ................................. l0 §2. The triple {R,~,~} on the category of spaces 13 §3. The total space of a cosimplicial space ...... 17 §4. The R-completion of a space .................. 20 §5. R-complete, R-good and R-bad spaces .......... 24 §6. Low dimensional behaviour .................... 30 §7. Disjoint unions, finite products and multiplicative structures .................... 34 §8. The fibre-wise R-completion .................. 40 §9. The role of the ring R ....................... 44 Chapter II. Fibre lemmas §i. Introduction ................................. 48 §2. The principal fibration lemma ................ 50 §3. Proof of the principal fibration lemma ....... 54 §4. Nilpotent fibrations ......................... 58 §5. The mod-R fibre lemma ........................ 62 §6. Proof of the mod-R fibre lemma ............... 67 Chapter III. Tower lemmas §i. Introduction ................................. 70 §2. Pro-isomorphisms of towers of groups ......... 73 §3. Weak pro-homotopy equivalences ............... 76 §4. Proof of 3.3 ................................. 79 §5. R-nilpotent spaces ........................... 82 §6. The tower lemmas ............................. 87 §7. Tower version of the mod-R fibre lemma ....... 91 §8. An Artin-Mazur-like interpretation of the R-completion ................................. 94 Chapter IV. An R-completion of ~roups and its relation to the R-completion of spaces §I. Introduction ................................. 99 §2. The R-completion of a group .................. 102 §3. A variation on the R-nilpotent tower lemma ... 106 §4. A relation between the "R-completion of a group" and the "R-completion of a space" . .... 109 §5. Applications ................................. 113 §6. Proof of proposition 4.3 ..................... 118 §7. Proof of proposition 4.2 (using 4.3) ......... 123 IV Chapter V. R-localizations of nilp0tent spaces §i. Introduction ................................. 126 §2. Malcev completions for nilpotent groups ...... 128 §3. Homotopy and homology properties of the R-completion of a nilpotent space (Re Q) ..... 133 §4. R-localizations of nilpotent spaces (Re Q) ... 137 §5. R-localizations of function spaces ........... 141 §6. Fracture lemmas .............................. 146 §7. Some slight generalizations .................. 152 §8. Fracturing H-space structures ................ 156 §9. Zabrodsky mixing of nilpotent spaces ......... 159 Chapter VI. p-completions of nilpotent spaces §l. Introduction ................................. 163 §2. Ext-completions and Hom-completions for nilpotent groups ............................. 165 §3. Ext-p-complete nilpotent groups .............. 172 §4. Examples of Ext and Hom completions .......... 179 §5. Homotopy and homology properties of the Zp-completion of a nilpotent space ........... 183 §6. p-completions of nilpotent spaces ............ 186 §7. p-completions of function spaces ............. 189 58. An arithmetic square fracture lemma .......... 192 §9. Curtis convergence theorems .................. 197 Chapter VII. A glimpse at the R-completi0n of non-nilpotent spaces §l. Introduction ................................. 202 52. Homotopy characterizations of the R-completion for R-good spaces ............... 204 §3. Spaces with an R-perfect fundamental group ... 206 §4. Spaces with finite homotopy or homology groups 211 §5. Spaces with a finite fundamental group ....... 215 §6. R-homotopy theories .......................... 218 Part II. Towers of fibrations, cosimplicial spaces and homotopy limits §0. Introduction to Part II ...................... 224 Chapter VIII. Simplicial sets and topological spaces §I. Introduction ................................. 228 §2. Simplicial sets .............................. 229 §3. Equivalence of simplicial and topological homotopy theories ............................ 237 §4. The homotopy relation and function spaces .... 244 V Chapter IX. Towers of fibrations §i. Introduction ................................. 249 §2. The functors lim and ~im I for groups ......... 251 §3. The homotopy groups of the inverse limit of a tower of fibrations ..................... 254 §4. The extended homotopy spectral sequence of a tower of fibrations ..................... 258 §5. Applications ................................. 261 Chapter X. Cosimplicial spaces §l. Introduction ................................. 265 §2. Cosimplicial spaces .......................... 267 §3. The total space of a cosimplicial space ...... 270 §4. Weak equivalences, cofibrations and fibrations 273 §5. Cosimplicial spaces form a closed simplicial model category ............................... 277 §6. The homotopy spectral sequence of a cosimplicial space ........................... 281 §7. A cosimplicial description of the E2-term .... 284 Chapter XI. Homotopy inverse limits §l. Introduction ................................. 287 §2. Some spaces associated with a small category . 291 §3. Homotopy inverse limits ...................... 295 §4. Examples and generalizations ................. 299 §5. Cosimplicial replacement of diagrams ......... 302 §6. The functors lim s for diagrams of (abelian) groups ............................. 305 §7. A spectral sequence for homotopy groups of homotopy inverse limit spaces ............. 309 §8. Homotopy inverse limits in terms of homotopy categories .......................... 313 §9. A cofinality theorem ......................... 316 §i0. Homotopy inverse limits for certain large diagrams of spaces ..................... 321 Chapter XII. Homotopy direct limits §i. Introduction ................................. 325 §2. Homotopy direct limits ....................... 327 §3. Examples and generalizations ................. 330 §4. A relation between homotopy direct and inverse limits ............................... 334 §5. Simplicial replacement of diagrams ........... 337 Bibliography ...................................... 341 Index ............................................. 345 Part I. Completions and localizations §0. Introduction to Part I Our main purpose in Part I of these notes, i.e. Chapters I through VII, is to develop for a ring R a functorial notion of R- completion of a s~ace X which (i) for R = Z (the integers modulo a prime p) and X P subject to the usual finiteness conditions, coincides, up to homoto~y, with the p-profinite completions of [Quillen (PG)] and [Sullivan, Ch.3], and (ii) for R ~ Q (i.e. R a subring of the rationals), coincides, up to homotopy, with the localizations of [Quillen (RH)], [Sullivan, Ch.2], [Mimura-Nishida-Toda] and others. Our R-completion is defined for arbitrary spaces, and throughout these notes we have tried to avoid unnecessary finiteness and simple connectivity assumptions. To develop our R-completion we need some homotopy theoretic results on towers of fibrations, cosimplicial spaces, and homotopy limits, which seem to be of interest in themselves and which we have therefore collected in Part II of these notes, i.e. Chapters VIII through XII. There are, we believe, two main uses for completions and localizations, i.e. for R-completions: first of all, they permit a "fracturing of ordinary homotopy theory into mod-p components"; and secondly, they can be used to construct important new (and old) spaces. Of course, the general idea of "fracturing" in homotopy theory is very old; and indeed, the habit of working mod-p or using Serre's 2 Part I, §0 C-theory is deeply ingrained in most algebraic topologists. However "fracturing" in its present form (due largely to Sullivan) goes considerably further and, among other things, helps explain the efficacy of the familiar mod-p methods. Roughly speaking (following Sullivan), one can use completions or localizations to "fracture" a homotopy type into "mod-p components" together with coherence information over the rationals; and the original homotopy type can then be recovered by using the coherence information to reassemble the "mod-p components". In practice the rational information often "takes care of itself", and ordinary homotopy theoretic problems (e.g. whether two maps are homotopic or whether a space admits an H-space structure) often reduce to "mod-p problems".
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