Graduate Texts in Mathematics 25 Editorial Board S Axler F.w Gehring P.R Halmos Springer-Science+Business Media, LLC Graduate Texts in Mathematics 10 II 12 I3 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 TAKEUTUZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces HILTON/STAMMBACH A Course in Homological Algebra 2nd ed MAC LANE Categories for the Working Mathematician HUGHES/PiPER Projective Planes SERRE A Course in Arithmetic TAKEUTIfZARlNG Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable l 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FULLER Rings and Categories of Modules 2nd ed GOLUBITSKy/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed HUSEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNES/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEwrrr/STRoMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARlsKiiSAMUEL Commutative Algebra VoU ZARISKUSAMUEL Commutative Algebra Vol.ll JACOBSON Lectures in Abstract Algebra l Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory of Fields and Galois Theory 33 HIRSCH Differential Topology 34 SPITZER Principles of Random Walk 2nd ed 35 WERMER Banach Algebras and Several Complex Variables 2nd ed 36 KELLEy/NAMIOKA et al Linear Topological Spaces 37 MONK Mathematical Logic 38 GRAUERT/FRlTZSCHE Several Complex Variables 39 ARVESON An Invitation to C*-Algebras 40 KEMENy/SNELL/KNAPP Denumerable Markov Chains 2nd ed 41 APOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed 42 SERRE Linear Representations of Finite Groups 43 GILLMAN/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LOEVE Probability Theory l 4th ed 46 LOEVE Probability Theory II 4th ed 47 MOISE Geometric Topology in Dimensions and 48 SACHS/WU General Relativity for Mathematicians 49 GRUENBERG/WEIR Linear Geometry 2nd ed 50 EDWARDS Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRAvERlWATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROWN/PEARCY Introduction to Operator Theory I: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CROWELL/Fox Introduction to Knot Theory 58 KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 2nd ed continued after index Graduate Texts in Mathematics 17O Editorial Board S Axler RW Gehring P.R Halmos Springer-Science+Business Media, LLC Graduate Texts in Mathematics T AKEliTriZARlNG Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces HILTON/STAMMBACH A Course in Homological Algebra MAC LANE Categories for the Working Mathematician HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic T AKEUTriZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory 10 COHEN A Course in Simple Homotopy Theory 11 CONWAY Functions of One Complex Variable I 2nd ed 12 BEALS Advanced Mathematical Analysis 13 ANDERSON/FuLLER Rings and Categories of Modules 2nd ed 14 GOLUBITSKy/GVILLEMIN Stable Mappings and Their Singularities 15 BERBERIAN Lectures in Functional Analysis and Operator Theory 16 WINTER The Structure of Fields 17 ROSENBLATT Random Processes 2nd ed 18 HALMOS Measure Theory 19 HALMOS A Hilbert Space Problem Book 2nd ed 20 HVSEMOLLER Fibre Bundles 3rd ed 21 HUMPHREYS Linear Algebraic Groups 22 BARNES/MACK An Algebraic Introduction to Mathematical Logic 23 GREVB Linear Algebra 4th ed 24 HOLMES Geometric Functional Analysis and Its Applications 25 HEWITT/STROMBERG Real and Abstract Analysis 26 MANES Algebraic Theories 27 KEl.LEY General Topology 28 ZARISKIISAMUEL Commutative Algebra VoU 29 ZARISKIISAMUEL Commutative Algebra VoU! 30 JACOBSON Lectures in Abstract Algebra ! Basic Concepts 31 JACOBSON Lectures in Abstract Algebra II Linear Algebra 32 JACOBSON Lectures in Abstract Algebra !II Theory of Fields and Gal'·is Theory 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 HIRSCH Differential Topology SPITZER Principles of Random Walk 2nd ed WERMER Banach Algebras and Several Complex Variables 2nd ed KELLEY/NAMIOKA et al Linear Topological Spaces MONK Mathematical Logic GRAUERT/FRlTZSCHE Several Complex Variables ARVESON An Invitation to CO-Algebras KEMENy/SNELL/KNAPP Denumerable Markov Chains 2nd ed ApOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed SERRE Linear Representations of Finite Groups GILLMAN/ JERISON Rings of Continuous Functions KENDIG Elementary Algebraic Geometry LOEVE Probability Theory I 4th ed LOEvE Probability Theory II 4th ed MOISE Geometric Topology in Dimensions and SACHS/WU General Relativity for Mathematicians GRUENBERG/WEIR Linear Geometry 2nd ed EDWARDS FeITnat's Last Theorem KLINGENBERG A Course in Differential Geometry HARTSHORNE Algebraic Geometry MANIN A Course in Mathematical Logic GRA VERlW ATKINS Combinatorics with Emphasis on the Theory of Graphs BROWN/PEARCY Introduction to Operator Theory I: Elements of Functional Analysis MASSEY Algebraic Topology: An Introduction CRowELL/Fox Introduction to Knot Theory KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed LANG Cyclotomic Fields ARNOLD Mathematical Methods in Classical Mechanics 2nd ed continued after index Glen E Bredon Sheaf Theory Second Edition i Springer Glen E Bredon Department of Mathematics Rutgers University New Brunswick, NJ 08903 USA Editorial Board S Axler Department of Mathematics Michigan State University East Lansing, MI 48824 USA F W Gehring Department of Mathematics University of Michigan Ann Arbor, MI 48109 USA P.R Halmos Department of Mathematics Santa Clara University Santa Clara, CA 95053 USA Mathematics Subject Classification (1991): 18F20, 32LlO, 54B40 Library of Congress Cataloging-in-Publieation Data Bredon, Glen E Sheaf theory / Glen E Bredon - 2nd ed p em - (Graduate texts in mathematics ; 170) lncludes bibliographical referenees and index ISBN 978-1-4612-6854-3 ISBN 978-1-4612-0647-7 (eBook) DOI 10.1007/978-1-4612-0647-7 Sheaf theory TitIe II Series QA612.36.B74 1997 514' 224-de20 96-44232 Printed on acid-free paper The first edition of this book was published by MeGraw Hill Book Co., New York - Toronto, Ont.-London, © 1967 © 1997 Springer Seience+Business Media New York Originally published by Springer-Verlag Berlin Heidelberg New York in 1997 Softcover reprint of the hardcover 2nd edition 1997 AII rights reserved This work may not be translated or eopied in whole or in part without the written permis sion of the publisher Springer-Seienee+Business Media, LLC, exeept for brief excerpts in connection with reviews or seholarly ana1ysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Produetion managed by Timothy Taylor; manufaeturing supervised by Jaequi Ashri Camera-ready eopy prepared from the author's LaTeX files 987654321 SPIN 10424939 Preface This book is primarily concerned with the study of cohomology theories of general topological spaces with "general coefficient systems." Sheaves play several roles in this study For example, they provide a suitable notion of "general coefficient systems." Moreover, they furnish us with a common method of defining various cohomology theories and of comparison between different cohomology theories The parts of the theory of sheaves covered here are those areas important to algebraic topology Sheaf theory is also important in other fields of mathematics, notably algebraic geometry, but that is outside the scope of the present book Thus a more descriptive title for this book might have been Algebraic Topology from the Point of View of Sheaf Theory Several innovations will be found in this book Notably, the concept of the "tautness" of a subspace (an adaptation of an analogous notion of Spanier to sheaf-theoretic cohomology) is introduced and exploited throughout the book The fact that sheaf-theoretic cohomology satisfies the homotopy property is proved for general topological spaces Also, relative cohomology is introduced into sheaf theory Concerning relative cohomology, it should be noted that sheaf-theoretic cohomology is usually considered as a "single space" theory This is not without reason, since cohomology relative to a closed subspace can be obtained by taking coefficients in a certain type of sheaf, while that relative to an open subspace (or, more generally, to a taut subspace) can be obtained by taking cohomology with respect to a special family of supports However, even in these cases, it is sometimes of notational advantage to have a relative cohomology theory For example, in our treatment of characteristic classes in Chapter IV the use of relative cohomology enables us to develop the theory in full generality and with relatively simple notation Our definition of relative cohomology in sheaf theory is the first fully satisfactory one to be given It is of interest to note that, unlike absolute cohomology, the relative cohomology groups are not the derived functors of the relative cohomology group in degree zero (but they usually are so in most cases of interest) The reader should be familiar with elementary homological algebra Specifically, he should be at home with the concepts of category and functor, with the algebraic theory of chain complexes, and with tensor products and direct limits A thorough background in algebraic topology is also necIThis is not even restricted to Hausdorff spaces This result was previously known only for paracompact spaces The proof uses the notion of a "relatively Hausdorff subspace" introduced here Although it might be thought that such generality is of no use, it (or rather its mother theorem II-I1.l) is employed to advantage when dealing with the derived functor of the inverse limit functor v vi Preface essary In Chapters IV, V and VI it is assumed that the reader is familiar with the theory of spectral sequences and specifically with the spectral sequence of a double complex In Appendix A we give an outline of this theory for the convenience of the reader and to fix our notation In Chapter I we give the basic definitions in sheaf theory, develop some basic properties, and discuss the various methods of constructing new sheaves out of old ones Chapter II, which is the backbone of the book, develops the sheaf-theoretic cohomology theory and many of its properties Chapter III is a short chapter in which we discuss the AlexanderSpanier, singular, de Rham, and Cech cohomology theories The methods of sheaf theory are used to prove the isomorphisms, under suitable restrictions, of these cohomology theories to sheaf-theoretic cohomology In particular, the de Rham theorem is discussed at some length Most of this chapter can be read after Section of Chapter II and all of it can be read after Section 12 of Chapter II In Chapter IV the theory of spectral sequences is applied to sheaf cohomology and the spectral sequences of Leray, Borel, Cartan, and Fary are derived Several applications of these spectral sequences are also discussed These results, particularly the Leray spectral sequence, are among the most important and useful areas of the theory of sheaves For example, in the theory of transformation groups the Leray spectral sequence of the map to the orbit space is of great interest, as are the Leray spectral sequences of some related mappings; see [15] Chapter V is an exposition of the homology theory of locally compact spaces with coefficients in a sheaf introduced by A Borel and J C Moore Several innovations are to be found in this chapter Notably, we give a definition, in full generality, of the homomorphism induced by a map of spaces, and a theorem of the Vietoris type is proved Several applications of the homology theory are discussed, notably the generalized Poincare duality theorem for which this homology theory was developed Other applications are found in the last few sections of this chapter Notably, three sections are devoted to a fairly complete discussion of generalized manifolds Because of the depth of our treatment of Borel-Moore homology, the first two sections of the chapter are devoted to technical development of some general concepts, such as the notion and simple properties of a cosheaf and of the operation of dualization between sheaves and cosheaves This development is not really needed for the definition of the homology theory in the third section, but is needed in the treatment of the deeper properties of the theory in later sections of the chapter For this reason, our development of the theory may seem a bit wordy and overcomplicated to the neophyte, in comparison to treatments with minimal depth In Chapter VI we investigate the theory of cosheaves (on general spaces) somewhat more deeply than in Chapter V This is applied to Cech homology, enabling us to obtain some uniqueness results not contained in those of Chaper V At the end of each chapter is a list of exercises by which the student Preface vii may check his understanding of the material The results of a few of the easier exercises are also used in the text Solutions to many of the exercises are given in Appendix B Those exercises having solutions in Appendix B are marked with the symbol @ The author owes an obvious debt to the book of Godement [40) and to the article of Grothendieck [41), as well as to numerous other works The book was born as a private set of lecture notes for a course in the theory of sheaves that the author gave at the University of California in the spring of 1964 Portions of the manuscript for the first edition were read by A Borel, M Herrera, and E Spanier, who made some useful suggestions Special thanks are owed to Per Holm, who read the entire manuscript of that edition and whose perceptive criticism led to several improvements This book was originally published by McGraw-Hill in 1967 For this second edition, it has been substantially rewritten with the addition of over eighty examples and of further explanatory material, and, of course, the correction of the few errors known to the author Some more recent discoveries have been incorporated, particularly in Sections II-16 and IV8 regarding cohomology dimension, in Chapter IV regarding the Oliver transfer and the Conner conjecture, and in Chapter V regarding generalized manifolds The Appendix B of solutions to selected exercises is also a new feature of this edition, one that should greatly aid the student in learning the theory of sheaves Exercises were chosen for solution on the basis of their difficulty, or because of an interesting solution, or because of the usage of the result in the main text Among the items added for this edition are new sections on tech cohomology, the Oliver transfer, intersection theory, generalized manifolds, locally homogeneous spaces, homological fibrations and p-adic transformation groups Also, Chapter VI on cosheaves and tech homology is new to this edition It is based on [12) Several of the added examples assume some items yet to be proved, such as the acyclicity of a contractible space or that sheaf cohomology and singular cohomology agree on nice spaces Disallowing such forward references would have impoverished our options for the examples As well as the common use of the symbol to signal the end, or absence, of a proof, we use the symbol to indicate the end of an example, although that is usually obvious Throughout the book the word "map" means a morphism in the particular category being discussed Thus for spaces "map" means "continuous function" and for groups "map" means "homomorphism." Occasionally we use the equal sign to mean a "canonical" isomorphism, perhaps not, strictly speaking, an equality The word "canonical" is often used for the concept for which the word "natural" was used before category theory gave that word a precise meaning That is, "canonical" certainly means natural when the latter has meaning, but it means more: that which might be termed "God-given." We shall make no attempt to define that concept precisely (Thanks to Dennis Sullivan for a theological discussion 490 Bibliography [75] Spanier, E H., Algebraic Topology, McGraw-Hill (1966) [76] Spanier, E H., Cohomology theory for general spaces, Annals of Math., 49 (1948) 407-427 [77] Specker, E., Additive Gruppen von Folgen ganzer Zahlen, Port Math., (1950) 131-140 [78] Steenrod, N., The Topology of Fibre Bundles, Princeton Univ Press (1951) [79] Steenrod, N., and Epstein, D B A., Cohomology Operations, Annals of Math Study, 50 (1962) [SO] Swan, R G., The Theory of Sheaves, Univ of Chicago Press (1964) [SI] Ungar, G S., Local homogeneity, Duke Math Jour., 34 (1967) 693-700 [S2] Whitehead, J H C., Note on the condition n-colc, Michigan Math Jour., (1957) 25-26 [S3] Wilder, R L., Monotone mappings of manifolds, I, Pacific J Math., (1957) 1519-152S; II, Michigan Math Jour., (1958) 19-23 [84] Wilder, R L., Some consequences of a method of proof of J H C Whitehead, Michigan Math Jour., (1957) 27-31 [S5] Wilder, R L., Topology of Manifolds, Amer Math Soc Colloquium Pub 32 (1949) [86] Yang, C T., p-adic transformation groups, Michigan Math Jour., (1960) 201-21S [S7] Young, G S., A characterization of 2-manifolds, Duke Math Jour., 14 (1947), 979-990 List of Symbols Ax 9'h-{(A) A/:::,.fiJ A(Y) AIY lsi fiJ/A ~ AA fA f*fiJ fiJ"-tA A0fiJ A*fiJ A0fiJ fifiJ AtJ)fiJ AxfiJ lin} A", :Jfom(A, fiJ) E( nY IY xW r w A*(XjG) AsH;P(XjA) S*(X;A) 9'*(XjA) ~H;P(XjA) O*(X) oH;P(X) 6;P(11; G) ~*(11; G) h;P(l1; G) 6;P(XjG) h;P(X;G) 3 7 10 11 11 12 12 14 18 18 19 19 19 19 2tl 21 22 22 22 22 23 23 23 23 24 25 26 26 26 27 27 27 27 28 28 28 covdimX Sn(X;A) ~*(X, A) 1* ;JtP(p*) HLC U + A ~*(X;A) l*(X;A) C;P(X;A) H;p(X;A) 3T*(X;A) F;(XjA) M*(XjA) Ext~,&l(A, fiJ) J(A) 9'*(XjA) $*(XjA) aUf3 axf3 f* ft N ~*(X,AjA) C;P(X, AjA) H;P(X, AjA) dim4>,L X dimLX Ind4> X indX H*(Xj L) clc2 Tp(A) St n , St j Sqj p1, DimX sH~(X;A) AH;p(X; fiJ) E~,q ==? Hp+q ( • ) U· 29 30 31 31 34 35 36 37 37 38 38 39 39 39 43 44 44 44 57 59 62 63 78 84 84 84 111 112 122 124 126 126 149 153 168 168 171 179 185 198 210 491 List of Symbols 492 w(U) h,Ji 1t'~ (j, jlA; Ji) cp(w) r dim~,LA 21(U) 6.(X; L) rcQ', rc{Q'} ~(21,M) 210 j21 B(J 21x 9' *(X, A; Ji) sH:(X,A;Ji) QlJ(21.; M) f)(21*; M) QlJ(Q'*; M) W*(XjJi) C~(XjJi) H:(XjJi) 1t'.(X;Ji) cp# C~(X,AjJi) H~(X,AjJi) W*(X,AjJi) 1t'; (f,fIA; Ji) n-whmL n-hmL @ ~ @-l O'.n(3 ~d(21,M) hld'], n-cmL 0'.-(3 0' (3 (a, (3) 2i Cp(U; 21) Hp(Uj 21) Hp(X;21) t,p(X; 21) E;,q(U) 210 210 213 219 222 237 281 281 282 285 286 286 286 287 288 289 290 290 292 292 292 293 299 305 305 305 322 329 329 329 329 331 336 349 350 375 345 345 346 419 424 424 425 425 432 ;{ > flabby; II-5.9 ;{ > cI>-soft '* I1> ;{ > cI>-softj II-Exercise 43 ;{ cI>(IIt)-soft '* f"iJ( ;{ cI>-softj IV-Exercise l ;{ cI>-soft '* ;{IA (cI>IA)-soft; II-9.2 ;{ cI>-soft '* ;{ A cI>-soft; II-9.13 lIJ (cI>IA)-soft '* ~ cI>-soft; II-9.12 E(cI» :) E(IIt), ;{ cI>-soft '* ;{ lIt-soft; II-16.5 (e) ;{ flabby (f) (g) (h) (i) (j) (k) (1) 493 494 List of Selected Facts (m) f!Il a sheaf of rings with unit, A an f!Il-module, f!Il => A ~-fine; II-9.16 ~-fine # f!Il ~-soft (n) A a ~o(X;L)-module => f.v.A a ~o(Y;L)-module; IV-3.4 (0) § injective => (p) A ~-fine (q) A ~-soft, (r) A> ~(.A,Đ) => A đ $ ~-soft flabby; II-Exercise 17 ~-finej II-9.18 torsion-free => A ® $ => li!J}.A> ~-soft; ~-soft; II-16.3I II-16.3D (s) A, $ c-fine, X locally compact, Hausdorff => A®$ c-fine; II-Exercise 14 (t) A, $ c-soft, A*$ = 0, X locally compact, Hausdorff => A®$ c-soft; II-Exercise 14 (u) Au ~-acyclic, all open U II-16.I (v) A ~-acyclic, all (w) A flabby, A c X (x) f: X -+ ~ # # A ~-soft # AIU (~IU)-acyclic, all U; A flabby; II-Exercise 2I ~-taut => AlA (~ n A)-acyclic; II-1O.5 Y locally c-Vietoris,.A c-fine on Y => f*.A c-fine; IV-6.7 Index A acyclic sheaf see sheaf, acyclic adjoint functor see functor, adjoint Alexander horned sphere, 385, 388 Alexander-Spanier sheaf see sheaf, Alexander-Spanier Alexander-Whitney formula, 25 Alexandroff, Po, 122, 379 augmentation, 34, 302 axioms Eilenberg-Steenrod, 83 for cohomology, 56 for cup product, 57 B base point nondegenerate, 95 Betti number local,379 Bing, R Ho, 392 Bockstein, 158 Boltjanskii, 471 Borel spectral sequence see spectral sequence, of Borel Borsuk, Ko, 392 bouquet, 130, 184, 277, 316 bundle fiber, 228 normal,256 orientable, 236 sphere, 234, 236, 254 universal, 246 vector, 233, 254 C Cantor set, 184, 228 cap product see product, cap carrier paracompact, 438 Cartan formula, 162, 168 Cartan spectral sequence see spectral sequence, of Cart an cis (cohomology fiber space), 395-397, 401, 416 chain locally finite, 206, 292 relative, 305 singular, 2, 31 change of rings, 195, 298, 333, 369, 372,443 Chern class, 254, 256 circle maps to, 96, 174 classifying space, 246 clc (cohomology locally connected), 126-127,129,131-133,139, 147,175,195,229,231-232, 277, 316, 351-352, 355, 357-358, 361, 363-366, 369, 379, 395, 414, 442-443 em (cohomology m.anifold), 375, 377, 383-384, 390, 392, 394, 399, 414 example of, 388 factorization of, 378 local orient ability of, 375 properties of, 380 with boundary, 384-388 coboundary, 25-26 cochain Alexander-Spanier, 24 de Rham, 27 singular, 2, 26-27 coefficients bundle of, 26 local, 26, 30 twisted,77 Cohen, Ho, 117, 123 cohomology Alexander-Spanier, 25, 29, 71, 185-186, 188, 206 at a point, 136 Cech, 27-29, 189-196 de Rham, 27, 7'1, 187, 194 finite generation of, 129 is not cohomology, 372 local, 136, 208 209 495 Index 496 of collapse, 90 of euclidean space, 82, 110 of infinite product, 130 of intersection, 208 of manifolds, 119 of product of spheres, 83 of product with a sphere, 83 of sphere, 82, 110 of suspension, 95 of union, 123, 209 reduced, 126 relative, 83, 86, 92, 182, 186, 206 sheaf theoretic, 38 single space, 86, 94 singular, 2, 26, 35, 71, 75, 83, 116, 179, 183, 186, 188, 195, 206, 276, 333 torsion-free, 172 with supports, 38 cohomology ring, 59 cohomomorphism, 14, 61 factorization of, 14 surjective, 84 comb space, 334 compact basewise, 23, 220 fiberwise, 23, 220 compactification end point, 391 one point, 133, 170 Stone-Cech, 173 cone, 95 conjunctive presheaf see presheaf, conjunctive Conner conjecture, 270, 274 Conner, P., 257, 267, 274, 276 continuity, 102-103, 315, 412-413 weak,74 coresolution, 432, 434, 437, 442, 445, 448 cosheaf,281 constant, 281 differential, 289, 431, 436, 445 differential, dual of, 289 direct image of, 286, 446 equivalent, 430 fhomomorphism of, 287 flabby, 281, 283, 285-287 singular, 281, 355, 443 supports in, 285 torsion-free, 283, 285, 287 weakly torsion-free, 412 covering acyclic, 193, 196, 278, 446 covering space see map, covering cross product see product, cross cup product see also product, cup associativity of, 58 commutativity of, 58 computation of, 60 CW-complex, 35, 183 D deck transformations, 251 Dedekind domain, 173 Deo, S., 171,463 derivative exterior, 27 derived functor see functor, derived diagonal class, 347 diagram, 16, 20 homotopically commutative, 175-176 differential, 25 total,60 differential form see form, differential differential sheaf see sheaf, differential Dim, 171, 462-463 dimension, 110-113, 115, 117-118, 121,125,170-171,177,209, 238-241, 257, 275, 383, 395 covering, 29, 122 decomposition theorem for, 241 inductive, 122, 124, 239 monotonicityof, 112-113 of connected space, 116 of dense set, 240 of homology manifold, 331 of manifold, 118 of product, 117-118, 239 of square, 178 of totally disconnected space, 116 of union, 123, 125, 241 497 Index product theorem for, 117-118, 239 relative, 237, 240-241, 275, 415 separation theorem for, 383-385 sum theorem for, 123, 125, 178, 208 weak inductive, 124 Zariski, 172 direct image see also image, direct derived functor of, 213 sections of, 31, 219 direct limit see limit, direct Dolbeault's theorem, 195 Dranishnikov, 122 duality Poincare, 77, 207, 330, 332, 341-342,354, 387 Whitney, 252 ductile map see map, ductile Dydak, J., 216, 242-243 E elementary sheaf see sheaf, elementary epiprecosheaf, 418 Euler characteristic, 143 Euler class, 236, 254, 276 exact locally, 420 exactness failure of, 10-11 excision, 83, 87, 89-90, 92, 307, 311-312 excisive see pair, excisive F family of supports, 21 dual of, 299, 413 extent of, 112 Fary spectral sequence see spectral sequence, of Fary Fary, I., 178 fiber restriction to, 214, 227 fiber bundle see bundle, fiber fibration spherical, 236, 254, 264-265, 275 figure eight, 326 fine see also sheaf, fine homotopically, 289 fixed point set, 143, 175, 257, 264, 276-277, 348, 407-411, 415 flabby see sheaf, flabby or cosheaf, flabby Floyd, E E., 143, 147, 257 form differential, 2, 27, 36, 69, 187, 354 function analytic, 5, 32 differentiable, 32 integer valued, 32 functor adjoint, 15 derived, 12, 43, 49, 53, 85 Hom, 9, 21, 31-32 of presheaves, 17 of sheaves, 17 functors connected sequence of, 18 G germ, 2, 32, 66, 69 see also sheaf, of germs Godement, 12,37, a9 group Lie, 178, 246, 267 transformation, 76, 80, 137, 175, 178, 194, 216, 246, 256-257, 264, 267, 270, 277, 327, 394, 397,399,403,407,413,415 Gysin sequence see sequence, Gysin H Hilbert cube, 118, 238, 328, 485 Hilbert space, 125 HLC (singular homology locally connected), 35, 71, 130-131, 180, 183-184, 186-187, 195, 232, 355-356, 358-359, 363, 413,441,444 semi-, 444-445 498 hle (homology locally connected), 350, 355, 439 semi-, 439-440, 448 hm (homology manifold), 329, 331, 342, 347, 373, 375, 380, 388-389, 392, 397, 409, 413, 415-416 dimension of, 331 homeomorphism local, 320 relative, 87-88, 183 homology Alexander-Spanier, 35 alternative theories, 371-372 Borel-Moore, 292, 439-440, 443, 446-447 Cech, 298, 315-316, 424-425, 435,438 de Rham, 36 in degree -1,293, 313 in degree 0, 314, 414 modified Borel-Moore, 442 reduced, 302 relative, 359 sequence, 39 single space, 313 singular, 7, 30, 32, 206, 281, 287-289, 321, 326, 332, 356, 358-359, 363, 413, 443, 445, 448 homology class fundamental, 338 homology sheaf see sheaf, homology homomorphism edge, 199-200, 202, 205, 223, 275, 277-278, 452 /-, of cosheaves, 287 /-, of sheaves, 411 induced, 56, 62, 64 of presheaves, of sheaves, homotopy, 79-80, 319 Hu, S T., 393 I ideal boundary, 135, 137 image, direct, 12, 49, 76, 170, 210, 276 inverse, 12 Index injective sheaf see sheaf, injective intersection number, 345 intersection product see product, intersection invariance of domain, 383 inverse image, 12 inverse limit see also limit, inverse derived functor of, 78, 208-209, 315-317 involution, 257, 276 isomorphism local, 13, 349, 421 isotropy subgroup, 247 J Jones, L., 145 Jussila, 0., 440 K kernel,9 Klein bottle, 327, 397 Knaster set, 117, 237, 240-241 Kiinneth theorem, 108, 109, 126, 175,231,227,339,364-367, 414 Kuz'minov, 119 L Lefschetz fixed point theorem, 348 lens space see space, lens Leray sheaf see sheaf, Leray Leray spectral sequence see spectral sequence, of Leray Lie group see group, Lie limit direct, 413, 20, 66-67, 102-103, 119, 169, 172, 174, 308-309, 412, 414 inverse, 100, 103, 171, 178, 225, 278, 298, 315-316, 427 Liseikin, 119 local homology group, 293 locally finite, 49 long line, 91, 122, 130, 169, 388, 460 Index M Mac Lane, S., 52 manifold, 129, 133-134, 137, 171, 207, 331-332 manifold cohomology, see em homology, see hm as identification space, 343 characterization of, 388 cohomology, 131 complex, 195 differentiable, 27, 36, 66, 69, 354 dimension of, 118 generalized, 373, 375, 377-378, 380, 383-384, 386-388, 392, 394 non-Hausdorff, singular homology, 332 map o-proper, 301-302 0- Vietoris, 225 covering, 76, 78, 143, 320, 333, 416 ductile, 195 equivariant, 178, 246, 248, 267 finite to one, 76, 225, 243-244 orbit, 264, 267,327,397-399,403 ell-closed, 74 proper, 80, 102 mapping cone, 157, 176 Mardesic, 438, 445 Mayer-Vietoris see sequence, Mayer-Vietoris minimality principle see principle, minimality Mittag-Leffler, 172,209,315,464 monodromy, 229 monopresheaf, 6, 23, 31 monotone mapping, 389, 397 Montgomery, D., 393 multifunctor, 53 N nerve, 29 Nobeling, G., 173, 293,314 Nunke, 373 o Oliver, R., 145, 267, 277, 476 operation 499 cohomology, 148 orbit, 137, 178, 216, 247, 267 orientable, 329 locally, 372, 377 orientation, 236, 268 orientation sheaf see sheaf, orientation P p-adic group, 394, 397, 399 classifying space of, 399 orbit space of, 398, 400 pair excisive, 96, 98-100, 170 pairing intersection, 345 Kronecker, 346 Poincare, 341-342 paracompact hereditarily, 21,66, 73, 112-113, 277 locally, 112 paracompactification, 170 paracompactifying, 22 for a pair, 70, 73 Poincare duality see duality, Poincare Poincare Lemma, 36 pointwise split, 18 Pol, E and R., 123 Pontryagin, 118 Pontryagin class, 254 precosheaf, 281 constant, 281 equivalent, 374, 422 local isomorphism of, 349, 421 locally constant, 374 locally zero, 349, 421, 426 smooth, 422, 428, 430, 448 presheaf, conjunctive, 6, 31 conjuntive, 22 constant, differential, 34 fcohomomorphism of, 14 graded,34 locally finitely generated, 217 mono-, principle minimality, 74, 169 Index 500 product cap, 336, 340-341, 343, 412-413 cartesian, 19 commutative, 177 cross, 59, 81, 100, 107, 126, 346 cup, 25, 27, 57, 64, 71, 81, 86, 94, 100, 149, 171, 178, 180-181, 183, 185-186,215, 225-226, 237 direct, 19 intersection, 344 tensor, 18, 70, 120 torsion, 18 total tensor, 19, 107, 110, 170, 175 total torsion, 19 projective plane, 216, 223 projective sheaf see sheaf, projective projective space, 13, 77 proper map see map, proper Q quasi-coresolution, 350, 352-353, 355 example of, 354-355 quasi-resolution, 350 quotient sheaf see sheaf, quotient R Raymond, F., 134, 137, 382, 391 reduced cohomology see cohomology, reduced reduced homology see homology, reduced refinement, 28 reflector, 422, 428, 430 relative homeomorphism see homeomorphism, relative relatively Hausdorff, 66, 73 resolution, 34, 47, 51, 111, 175 canonical, 37, 39 canonical injective, 44 homotopically trivial, 37-38 injective, 42, 45 restriction, 73 Roy, P., 125 Rubin, L R., 241 S 81, 82, section, 2, discontinuous, 16, 36 locally trivial, of homology sheaf, 326 presheaf of, support of, zero, sections restriction of, 24 sequence admissible, 52 cohomology, 84, 170 connected, 18, 52, 56 exact, fundamental, 53 Gysin, 236 homology, 292, 305,307 left exact, 9, 22 locally exact, 411 Mayer-Vietoris, 94, 98, 100, 169, 412 of a pair, 72 of order two, 34 of triple, 88 pointwise split, 18, 52 Smith, 142, 408 8mith-Gysin, 251, 264-265 Wang, 237 Kiinneth, see Kiinneth theorem, serration, 16, 36 sheaf,3 acyclic, 46-47, 49, 65, 68, 110-111,171,175,276 Alexander-Spanier, 201 concentrated on subspace, 106-107, 174, 212, 238 constant, 7, 31 constant, subsheaf of, 12, 32 derived, 34, 174 differential, 34, 198, 202 elementary, 295, 297, 305, 307-309,311-313,325 J-cohomomorphism of, 14 J-homomorphism of, 411 filtered differential, 257 fine, 69-70, 107, 169-170,225, 276 Index flabby, 47, 49, 66, 70-71, 170-172, 175, 294, 296-298 fundamental theorem of, 202, 205 generated, 3, 5, graded, 34 Hom, 21 homology, 7, 34, 137, 322 homotopically fine, 172 injective, 41, 48, 170,293 Leray, 213-216, 218, 222, 227-229, 231-233, 236, 252, 267, 277, 395 Leray, of bundle, 228 Leray, of projection, 227, 229 locally constant, 7, 31, 178 locally finitely generated, 217 of germs, 3, 10, 174, 195 of local homology groups, 293 of modules, of rings, 3, 8, 38, 211 orientation, 7, 77, 207, 234, 329, 386, 414 orientation, of product, 378 projective, 30 quotient, 10, 16 replete, 293, 295, 297, 301, 308-309, 403 restriction of, 7, 66, 71 soft, 65-66, 68-70, 102, 110-111, 119-120,170,175,210,275, 282, 284, 286, 288, 299, 352, 403, 464 topology of, 2, 4-5 torsion-free, 38, 177, 286 twisted, 13 weakly torsion-free, 412 sine curve topologist's, 74, 184, 277 Skljarenko, E G., 121, 232, 238, 245-246 slice, 178 Smith sequence see sequence, Smith Smith theorem, 143,409 Smith, P A., 143 Smith-Gysin sequence see sequence, Smith-Gysin soft see sheaf, soft 501 solenoid, 80, 103, 228, 315, 332-333, 367, 372-373, 397 embedding of, 333 space acyclic, 79-80, 144, 147, 169, 271-274, 319 countable connected, 125 lens, 145, 472 locally connected, 126, 133-134 locally contractible, 35, 131 locally homogeneous, 392 locally isotopic, 393 orbit, 137, 144, 146-147, 216, 271,398, 400, 409 paracompact, 21 rudimentary, 78, 178, 317 simply ordered, 169 well pointed, 95 zero dimensional, 122 Spanier, E H., 35, 73 Specker, E., 173 spectral sequence for inverse system, 207 for relative cohomology, 206 for singular cohomology, 276 for singular homology, 206 of a cIs, 396, 484 of a covering, 278, 431-432, 436 of Borel, 248, 275 of Cartan, 251 of Fary, 263-265 of filtered differential sheaf, 258, 262 of filtered space, 263 of inclusion, 240 of Leray, 222, 225, 229-230, 233, 236-237, 263, 268, 270, 275 of map for homology, 324 sphere separation of, 275 sphere bundle see bundle, sphere stalk, 3, Steenrod, 30, 148, 162 Steenrod power, 168 Steenrod square, 168 Stiefel-Whitney class, 254, 276 Stokes' theorem, 188 structure group, 228 subdivision, 26 Index 502 subsheaf,9 subspace locally closed, 11 neighborhoods of, 73 taut, see taut sum direct, 19 support, 7, 281 supports closed,22 compact, 22, 74 empty, 22 extension of, 219 extent of, 22 family of, 21, 23, 219 suspension, 95 system direct, 20, 30 inverse, 100 T taut, 35, 73, 85, 90-91, 99, 169-170 hereditarily, 169 tensor product universal property of, 31 theory cohomology, 56 Thorn class, 235, 252, 276 Thorn isomorphism, 235, 276 Torhorst, 134 torsion, 172-173 transfer, 77, 139, 321, 405 of Oliver, 268-270 transformation group see group, transformation tube, 178, 267 Tychonoff plank, 467 U Ungar, G S., 393 unity partition of, 170 universal coefficient theorem, 109, 275, 298, 352, 363, 365, 414 V vector bundle see bundle, vector Vietoris, 76, 78, 222, 245-246, 317 Vietoris map, 323, 389-390, 398 C-, 225 W Walsh, J., 216, 242-243 Wang sequence see sequence, Wang Whitehead, J H C., 379 Whitney duality see duality, Whitney whm (weak homology manifold), 329-330, 333, 342, 384, 413, 416 wild arc, 390 Wilder's necklace, 131-132, 278 Wilder, R L., 127, 131, 133-134, 379,389 y Young, G S., 388 Z Zariski space, 172 zero extension by, 11, 49, 71, 212, 286 locally, 9, 349 semilocally, 420 Graduate Texts in Mathematics continued from page ii 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 WHITEHEAD Elements of Homotopy Theory KARGAPOLOV/MERLZJAKOV Fundamentals of the Theory of Groups BOLLOBAS Graph Theory EDWARDS Fourier Series Vol I 2nd ed WELLS Differential Analysis on Complex Manifolds 2nd ed WATERHOUSE Introduction to Affine Group Schemes SERRE Local Fields WEIDMANN Linear Operators in Hilbert Spaces LANG Cyclotomic Fields II MASSEY Singular Homology Theory FARKAS/KRA Riemann Surfaces 2nd ed STILLWELL Classical Topology and Combinatorial Group Theory 2nd ed HUNGERFORD Algebra DAVENPORT Multiplicative Number Theory 2nd ed HocHSCHILD Basic Theory of Algebraic Groups and Lie Algebras lITAKA Algebraic Geometry HECKE Lectures on the Theory of Algebraic Numbers BURRIS/SANKAPPANAVAR A Course in Universal Algebra WALTERS An Introduction to Ergodic Theory ROBINSON A Course in the Theory of Groups 2nd ed FORSTER Lectures on Riemann Surfaces BOTT/Tu Differential Forms in Algebraic Topology WASHINGTON Introduction to Cyclotomic Fields 2nd ed IRELAND/RoSEN A Classical Introduction to Modern Number Theory 2nd ed EDWARDS Fourier Series Vol II 2nd ed VAN LINT Introduction to Coding Theory 2nd ed BROWN Cohomology of Groups PIERCE Associative Algebras LANG Introduction to Algebraic and Abelian Functions 2nd ed BRONDSTED AI' Introduction to Convex Polytopes BEARDON On the Geometry of Discrete Groups 92 DIESTEL Sequences and Series in Banach 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 III 112 113 114 lIS 116 117 118 Spaces DUBROVIN/FoMENKO/NoVIKOV Modem Geometry-Methods and Applications Part I 2nd ed WARNER Foundations of Differentiable Manifolds and Lie Groups SHIRYAEV Probability 2nd ed CONWAY A Course in Functional Analysis 2nd ed KOBLITZ Introduction to Elliptic Curves and Modular Forms 2nd ed BROCKERIToM DIECK Representations of Compact Lie Groups GRovE/BENSON Finite Reflection Groups 2nd ed BERG/CHRISTENSEN/RESSEL Harmonic Analysis on Semi groups: Theory of Positive Definite and Related Functions EDWARDS Galois Theory VARADARAJAN Lie Groups, Lie Algebras and Their Representations LANG Complex Analysis 3rd ed DUBROVIN/FoMENKO/N OVIKOV Modem Geometry-Methods and Applications Part II LANG SL,(R) SILVERMAN The Arithmetic of Elliptic Curves OLVER Applications of Lie Groups to Differential Equations 2nd ed RANGE Holomorphic Functions and Integral Representations in Several Complex Variables LEHTO Univalent Functions and Teichmiiller Spaces LANG Algebraic Number Theory HUSEMOLLER Elliptic Curves LANG Elliptic Functions KARATZAS/SHREVE Brownian Motion and Stochastic Calculus 2nd ed KOBLITZ A Course in Number Theory and Cryptography 2nd ed BERGERIGOSTIAUX Differential Geometry: Manifolds, Curves, and Surfaces KELLEy/SRINIVASAN Measure and Integral Vol I SERRE Algebraic Groups and Class Fields PEDERSEN Analysis Now 119 ROTMAN An Introduction to Algebraic Topology 120 ZIEMER Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation 121 LANG Cyclotomic Fields I and II Combined 2nd ed 122 REMMERT Theory of Complex Functions Readings in Mathematics 123 EBBINGHAUS/HERMES et al Numbers Readings in Mathematics 124 DUBROVIN/FoMENKO/NoVIKOV Modem Geometry-Methods and Applications Part III 125 BERENSTEIN/GAY Complex Variables: An Introduction 126 BOREL Linear Algebraic Groups 127 MASSEY A Basic Course in Algebraic Topology 128 RAUCH Partial Differential Equations 129 FULTON/HARRlS Representation Theory: A First Course Readings in Mathematics 130 DODSON/POSTON Tensor Geometry 131 LAM A First Course in Noncommutative Rings 132 BEARDON Iteration of Rational Functions 133 HARRlS Algebraic Geometry: A First Course 134 ROMAN Coding and Information Theory 135 ROMAN Advanced Linear Algebra 136 ADKINS/WEINTRAUB Algebra: An Approach via Module Theory 137 AXLER/BoURDON/RAMEY Harmonic Function Theory 138 COHEN A Course in Computational Algebraic Number Theory 139 BREDON Topology and Geometry 140 AUBIN Optima and Equilibria An Introduction to Nonlinear Analysis 141 BECKER/WEISPFENNING/KREDEL Grobner Bases A Computational Approach to Commutative Algebra 142 LANG Real and Functional Analysis 3rd ed 143 DOOB Measure Theory 144 DENNIS/FARB Noncommutative Algebra 145 VICK Homology Theory An Introduction to Algebraic Topology 2nd ed 146 BRIDGES Computability: A Mathematical Sketchbook 147 ROSENBERG Algebraic K -Theory and Its Applications 148 ROTMAN An Introduction to the Theory of Groups 4th ed 149 RATCLIFFE Foundations of Hyperbolic Manifolds 150 EISENBUD Commutative Algebra with a View Toward Algebraic Geometry lSI SILVERMAN Advanced Topics in the Arithmetic of Elliptic Curves 152 ZIEGLER Lectures on Polytopes 153 FuLTON Algebraic Topology: A First Course 154 BROWN/PEARCY An Introduction to Analysis 155 KASSEL Quantum Groups 156 KECHRIS Classical Descriptive Set Theory 157 MALLIA VIN Integration and Probability 158 ROMAN Field Theory 159 CONWAY Functions of One Complex Variable II 160 LANG Differential and Riemannian Manifolds 161 BORWEIN/ERDEL vI Polynomials and Polynomial Inequalities 162 ALPERIN/BELL Groups and Representations 163 DIXON/MORTIMER Permutation Groups 164 NATHANSON Additive Number Theory: The Classical Bases 165 NATHANSON Additive Number Theory: Inverse Problems and the Geometry of Sumsets 166 SHARPE Differential Geometry: Cartan' s Generalization of Klein's Erlangen Program 167 MORANDI Field and Galois Theory 168 EWALD Combinatorial Convexity and Algebraic Geometry 169 BHATIA Matrix Analysis 170 BREDON Sheaf Theory 2nd ed ... loss of generality since we are interested here in local matters) We have the differential presheaf (1) where G > SO(U; G) is the usual augmentation Here we regard G as the constant presheaf U... sheaf J1 is called the sheaf generated by the presheaf A As we have noted, this is denoted by J1 = 9'heo{(A) or J1 = 9'heo{(U f-+ A(U» I Sheaves and Presheaves 1.6 Let u1 be a sheaf, A the presheaf... and Ker g' are generated respectively by the presheaves Imf and Ker g Similarly, the sheaf Ker g' / 1m f' is (naturally isomorphic to) the sheaf generated by the presheaf Kerg/lmf: U f > Kerg(U)/Imf(U)