Graduate Texts in Mathematics 20 Managing Editors: P R Halmos C C Moore Dale Husemoller Fibre Bundles Second Edition Springer-Verlag Berlin Heidelberg GmbH Dale Husemoller Haverford College Department of Mathematics Haverford, Pennsylvania 19041 Managing Editors P R Halmos C C Moore Indiana University Department of Mathematics Swain Hall East Bloomington, Indiana 47401 University of California at Berkeley Department of Mathematics Berkeley, California 94720 AMS Subject Classifications 14F05, 14F15, 18F15, 18F25, 55FXX Library of Congress Cataloging in Publication Data Husemoller, Dale Fibre bundles (Graduate texts in mathematics; v 20) Bibliography: p 313 Includes index Fibre bundles (Mathematics) I Title II Series QA612.6.H87 1975 514' 224 74-23157 All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag © 1966 by Dale Hausemoller Originally published by Springer-Verlag Berlin Heidelberg New York 1996 Softcover reprint of the hardcover 2nd edition in 1996 ISBN 978-1-4757-4010-3 ISBN 978-1-4757-4008-0 (eBook) DOI 10.1007/978-1-4757-4008-0 TO MY MOTHER AND THE MEMORY OF MY FATHER PREFACE The notion of a fibre bundle first arose out of questions posed in the 1930s on the topology and geometry of manifolds By the year 1950 the definition of fibre bundle had been clearly formulated, the homotopy classification of fibre bundles achieved, and the theory of characteristic classes of fibre bundles developed by several mathematicians, Chern, Pontrjagin, Stiefel, and Whitney Steenrod's book, which appeared in 1950, gave a coherent treatment of the subject up to that time About 1955 Milnor gave a construction of a universal fibre bundle for any topological group This construction is also included in Part I along with an elementary proof that the bundle is universal During the five years from 1950 to 1955, Hirzebruch clarified the notion of characteristic class and used it to prove a general Riemann-Roch theorem for algebraic varieties This was published in his Ergebnisse Monograph A systematic development of characteristic classes and their applications to manifolds is given in Part III and is based on the approach of Hirzebruch as modified by Grothendieck In the early 1960s, following lines of thought in the work of A Grothendieck, Atiyah and Hirzebruch developed K-theory, which is a generalized cohomology theory defined by using stability classes of vector bundles The Bott periodicity theorem was interpreted as a theorem in K-theory, and J F Adams was able to solve the vector field problem for spheres, using K-theory In Part II an introduction to K-theory is presented, the nonexistence of elements of Hopf invariant proved (after a proof of Atiyah), and the proof of the vector field problem sketched I wish to express gratitude to S Eilenberg, who gave me so much encouragement during recent years, and to J C Moore, who read parts of vii viii Preface the manuscript and made many useful comments Conversations with J F Adams, R Bott, A Dold, and F Hirzebruch helped to sharpen many parts of the manuscript During the writing of this book, I was particularly influenced by the Princeton notes of J Milnor and the lectures of F Hirzebruch at the 1963 Summer Institute of the American Mathematical Society Dale Husemoller PREFACE TO THE SECOND EDITION In this edition we have added a section to Chapter 15 on the Adams conjecture and a second appendix on the suspension theorems For the second appendix the author profited from discussion with Professors Moore, Stasheff, and Toda I wish to express my gratitude to the following people who supplied me with lists of corrections to the first edition: P T Chusch, Rudolf Fritsch, David C Johnson, George Lusztig, Claude Schocket, and Robert Sturg Part of the revision was made while the author was a guest of the I.H.E.S in January, May, and June 1974 Dale Husemoller CONTENTS PREFACE vii PRELIMINARIES ON HOMOTOPY THEORY 1 Category theory and homotopy theory Complexes The spaces Map (X, Y) and Mapo (X, y) 4 Homotopy groups of spaces Fibre maps PART I THE GENERAL THEORY OF FIBRE BUNDLES GENERALITIES ON BUNDLES 11 Definition of bundles and cross sections 11 Examples of bundles and cross sections 12 Morphisms of bundles 14 Products and fibre products 15 Restrictions of bundles and induced bundles Local properties of bundles 20 Prolongation of cross sections 20 Exercises 22 17 VECTOR BUNDLES 23 Definition and examples of vector bundles 23 Morphisms of vector bundles 25 i% x Contents Induced vector bundles 26 Homotopy properties of vector bundles 27 Construction of Gauss maps 29 Homotopies of Gauss maps 31 Functorial description of the homotopy classification of vector bundles Kernel, image, and cokernel of morphisms with constant rank 34 Riemannian and Hermitian metrics on vector bundles 35 Exercises 37 32 GENERAL FIBRE BUNDLES 10 11 12 13 39 Bundles defined by transformation groups 3\1 Definition and examples of principal bundles 40 Categories of principal bundles 41 Induced bundles of principal bundles 42 Definition of fibre bundles 43 Functorial properties of fibre bundles 45 Trivial and locally trivial fibre bundles 46 Description of cross sections of a fibre bundle 46 Numerable principal bundles over B X [0, 1] 48 The cofunctor kG 51 The Milnor construction 52 Homotopy classification of numerable principal G-bundles 54 Homotopy classification of principal G- bundles over CW- complexes Exercises 57 LOCAL COORDINATE DESCRIPTION OF FIBRE BUNDLES 59 Automorphisms of trivial fibre bundles 59 Charts and transition functions 60 Construction of bundles with given transition functions Transition functions and induced bundles 63 Local representation of vector bundle morphisms 64 Operations on vector bundles 65 Transition functions for bundles with metrics 67 Exercises 69 62 CHANGE OF STRUCTURE GROUP IN FIBRF BUNDLES 70 Fibre bundles with homogeneous spaces as fibres 70 Prolongation and restriction of principal bundles 71 Restriction and prolongation of structure group for fibre bundles Local coordinate description of change of structure group 73 72 57 Contents Classifying spaces and the reduction of structure group 73 Exercises 74 CALCULATIONS INVOLVING THE CLASSICAL GROUPS 75 10 11 12 Stiefel varieties and the classical groups 75 Grassmann manifolds and the classical groups 78 Local triviality of projections from Stiefel varieties 79 Stability of the homotopy groups of the classical groups 82 Vanishing of lower homotopy groups of Stiefel varieties 83 Universal bundles and classifying spaces for the classical groups 83 Universal vector bundles 84 Description of all locally trivial fibre bundles over suspensions 85 Characteristic map of the tangent bundle over S" 86 Homotopy properties of characteristic maps 89 Homotopy groups of Stiefel varieties 91 Some of the homotopy groups of the classical groups 92 Exercises 95 PART II ELEMENTS OF K-THEORY STABILITY PROPERTIES OF VECTOR BUNDLES 99 Trivial summands of vector bundles 99 Homotopy classification and Whitney sums 100 The K cofunctors 102 Corepresentations of K, 105 Homotopy groups of classical groups and K,(S') 108 Exercises 109 RELATIVE K-THEORY 110 Collapsing of trivialized bundles 110 Exact sequences in relative K-theory 112 Products in K-theory 116 The cofunctor L(X, A) 117 The difference morphism 119 Products in L(X, A) 121 The clutching construction 122 The cofunctor L,.(X, A) 124 Half-exact cofunctors 126 Exercises 127 lei sii Contents 10 BOTT PERIODICITY IN THE COMPLEX CASE 128 K-theory interpretation of the periodicity result 128 Complex vector bundles over X X 82 129 Analysis of polynomial clutching maps 131 Analysis of linear clutching maps 133 The inverse to the periodicity isomorphism 136 11 CLIFFORD ALGEBRAS 139 Unit tangent vector fields on spheres: I 139 Orthogonal mUltiplications 140 Generalities on quadratic forms 141 Clifford algebra of a quadratic form 144 Calculations of Clifford algebras 146 Clifford modules 149 Tensor products of Clifford modules 154 Unit tangent vector fields on spheres: II 156 The group Spin(k) 157 Exercises 158 12 THE ADAMS OPERATIONSi AND REPRESENTATIONS 10 11 159 X-rings 159 The Adams ,y-operations in X-ring 160 The -y' operations 162 Generalities on G-modules 163 The representation ring of a group G and vector bundles 165 Semisimplicity of G-modules over compact groups 166 Characters and the structure of the group R,(G) 168 Maximal tori 169 The representation ring of a torus 172 The ,y-operations on K(X) and KO(X) 173 The ,y-operations on i(Sn) 175 13 REPRESENTATION RINGS OF CLASSICAL GROUPS Symmetric functions 176 Maximal tori in SU(n) and U(n) 178 The representation rings of SU(n) and U(n) Maximal tori in Sp(n) 180 Formal identities in polynomial rings 180 The representation ring of Sp(n) 181 179 176 On the Double Suspension 311 8.2 Remark Using the decomposition of snSn+l given in (3.3), we have the following description of [QSn+t, nSn+lj [nSn+t, nSn+lj = [SnSn+t, Sn+1j = [V l;;;i = llt;;;i 'lrni+1(Sn+l) Sni+1, Sn+1j Hence the class of up:ns2np-1 -+ nS 2np- is a sequence of elements in which for i = is just the element of degree p in 'lr2np_1(S2n p -1) Toda asserts in [1956, (8.3)] that all the components of Up in 'lr2(np_1)i+l(S2np-1) for i > are zero In other words Up = nv p where Vp: S2np-1 -+ S2np-1 is of degree p, that is, Up deloops 'lr2(np_1)i+1(S2n p -1) BIBLIOGRAPHY Adams, J F.: On the non-existence of elements of Hopf invariant one, Bull Am Math Soc., 64: 279-282 (1958) On the structure and applications of the Steenrod algebra, Comment, Math Helv., 32: 180-214 (1958) On the non-existence of elements of Hopf invariant one, Ann Math., 72: 20-104 (1960) On Chern characters and the structure of the unitary group, Proc Cambridge Phil Soc., 57: 189-199 (1961) Vector-fields on spheres, Bull Am Math Soc., 68: 39-41 (1962) Vector fields on spheres, Ann Math., 75: 603-632 (1962) On the groups J(X) I, II, 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charts, for fibre bundle, 60 for manifold, 248 Base space, 11 Bilinear form, symmetric, 142 Bott periodicity, 128, 137, 222 and integrality theorem, 279, 280 Bundle(s), 11 collapsed, 110 fibre [see Fibre bundle(s)] fibre product of, 15 G-,40 Hopf, 129 induced, 18 locally isomorphic, 20 locally trivial, 20 morphism of, 14 normal (see Normal bundle) principal [see Principal bundle(s)] product of, 15 restricted, 17 Bundle(s), sub-,ll tangent (see Tangent bundle) Bundle(s), universal (see Universal bundle) vector [see Vector bundle(s)] Character ring, 168 representation ring as, 168, 169 Characteristic classes, 267, 268 calculations, on canonical line bundles, 235, 244-246 on tangent bundles of Sn, Rpn, and CP', 237, 279 Chern (see Chern classes) complex, 269-271 in dimension n, 268, 269 Euler (see Euler class) Pontrjagin (see Pontrjagin classes) real, mod 2, 271, 272 2-divisible, 273-276 and representations, 281-283 Stiefel-Whitney (see Stiefel-Whitney classes) Characteristic map, 87-91 Charts, of fibre bundles, 60 of manifolds, 248 of vector bundles, 23, 60 Chern classes, 234, 268-271 axiomatic properties of, 234 definition, 234 multiplicative property of, 238, 239 323 324 Index Classical groups, 75 classifYing spaces for, 83 examples, 78 homotopy groups of, 82, 92-95 infinite, 76 stability of, 82 universal bundle for, 83 Classifying spaces, 52 for classical groups, 83 cohomology of, 269, 272, 274 of reduced structure group, 73-74 of vector bundles, 33, 84 Clifford algebras, 144 calculations of, 146 table, 148 Clifford modules, 149 table, 1.50 tensor products of, 1.54 Clutching construction, 123 Clutching maps, 123 Laurent, 130 linear, 133-136 polynomial, 131, 132 Cobordism, 262, 263 Cofunctor, half-exact (see Half-exact cofunctor) coH-space, Cokernel of morphism, 34 Collapsed bundle, 110 Compact group, 166 maximal tori of, 169-172 rank of, 171 representation ring of, 166-169 Weyl group of, 172 Compact-open topology, Cone over a space, Coreducible spaces, 214 relation to vector fields, 218-219 Cross section(s), 11 and Euler class, 242 of fibre bundles, 46 prolongation of, 21 CW-complex(es), homotopy classification over, 57 Difference isomorphism, 118 Duality, in manifolds, 256, 257 Duality theorem, Poincare, 2.57 Euclidean inner product, 12 Euclidean norm, 12 Euclidean space, orientation in, 2.51-253 Euler characteristic, definition, 2.59 and Euler class, 260 of stable vector bundles, 125 and vector fields, 260 Euler class, 240, 273-277 and cross sections, 242 definition, 240 and Euler characteristic, 260 of a manifold, 260 multiplicative property, 241, 242 and Thom isomorphism, 241, 244 Fibre, 11 of a fibre bundle, 44 of a principal bundle, 41 Fibre bundle(s), 44 atlas of charts for, 60 automorphisms, 59 classification of, 54-.'i7 cross section of, 46 locally trivial, 46 morphism of, 45 over suspension, 85 trivial, 46 Fibre homotopy equivalence, 208-210, 285 Fibre homotopy type, 208-210, 28.5 stable, 208-213 and Thom spaces, 213 Fibre maps, 7, 285 Fibre product, 15 Gauss map, 29 G-module, 163 direct sum of, 164 exterior product of, 164 morphism of, 164 semisimple, 165, 167 tensor product of, 164 Grassman manifold (or variety), 13,24,33 cohomology of, 269, 272, 274 as homogeneous space, 78, 79 Group(s), classical (see Classical groups) compact (see Compact group) homotopy (see Homotopy groups) Group(s), linear, 39 Index Group(s), orthogonal (see Orthogonal group) reduction of structure, 73-74 spin (Spin(n» (see Spin group) symplectic (see Symplcctic group) topological, 39 transformation, 39 unitary (see Unitary group) G-space,39 morphism of, 40 principal, 41 Gysin sequence, 240 Half-exact cofunctor, 126 Puppe sequence of, 127 Hermitian metrics of vector bundle, 36 Homotopy classification, over CWcomplexes, ';7 of principal bundles, 1)4-57 of vector bundle, 27-33, 101 Homotopy equivalence, fibre, 208-210, 28.5 Homotopy groups, of classical groups, 82, 92-9.5 of O(n), 82, 92-9.5 of SO(n), 82, 92-91) of Sp(n), 82, 92-9,'> of Stiefel variety, 83, 91 of SU(n), 82, 92-95 of U (n), 82, 92-95 Homotopy type, fibre (see Fibre homotopy type) Hopf bundle, 129 Hopf invariant, 196-202, 280, 281, 2BS-301) H-space, Intcgrality theorem of Bott, 279, 280 J(X),21O calculation of J(RPn), 223-225 of J(Sk), 211-213 K-cup product, 116 k-space, K(X) and KO(X), 103, 109 Bott periodicity of, 128, 137, 222 as A-ring, 159, 173-171) and representation ring, 166 as a ring, 116 325 K(X) and £O(X), 104, 109 calculation of K(Sn), 138 core presentation of, 107 table of results for KO(Sn) and KO(Rpn), 222, 223 A-ring (s), 159 Adams operations in, 160 K(X) and KO(X) as, 159, 173-175 -y-operations in, 162 representation ring as, 166, 173 split, 161 Leray-Hirsch theorem, 231 Linear groups, 39 Loop space, Manifold(s), 248-26,'> atlas of charts for, 248 duality in, 21)6, 257 Euler class of, 260 fundamental class of, 2.'')8 Grassman [see Grassman manifold (or variety)] orientation of, 253-255 Stiefel-Whitney classes of, 261 tangent bundle to, 250 Thorn class of, 258 Map(s), clutching (see Clutching maps) fibre, 7, 285 Gauss, 29 normal bundle of, 251 splitting, 237 Map space, Mapping cone, 113 Mapping cylinder, 113 Maximal tori, of compact groups, 169-172 of SO(n), 182 of Sp(n), 180 of Spin(n), 183 of SU(n), 178 of U(n), 178 Mayer-Vietoris sequence, 232, 21)4, 256 Milnor's construction of a universal bundle, 52-54 verification of universal property, 55-57 326 lnde% Morphism, B-, 14 of bundles, 14 cokernel of, 34 of fibre bundles, 45 of G-module, 164 of G-space, 40 image of, 34 kernel of, 34 local representation of, 64 of principal bundles, 41 of vector bundles, 25 Normal bundle, of an immersion, 263 of a map, 251 to the sphere, 12 Numerable covering, 48, 284 Orientation, in euclidean space, 251-253 of manifolds, 253-255 of vector bundles, 246 Orientation class, 253, 254 Orthogonal group (O(n» and (SO(n», 39,75 examples, 78 homotopy groups of, 82, 92-95 infinite, 76 maximal tori of, 182 representation ring of, 187-190 Weyl group of, 182 Orthogonal multiplication, 140 Orthogonal splitting, 142, 143 Path space, Periodicity, Bott (see Bott periodicity) Poincare duality theorem, 257 Pontrjagin classes, 245, 273-277 Principal bundle(s), 41 homotopy classification of, 54-57 induced,42 morphism of, 41 numerable, 48 Products, of bundles, 15 euclidean inner, 12 of G-module, exterior, 164 tensor, 164 reduced,5 Projection, 11 Projective space, tangent bundle of, 13, 17, 237 Puppe sequence, 112-115, 127, 206 Quadratic form, 142 Rank of a compact group, 171 Reduced product, Reducible spaces, 214 relation to vector fields, 216-218 Representation(s), 163-165 and characteristic classes, 281-283 local, of morphism, 64 real, of Spin(n), 193-195 semisimple, 165, 167 spin (see Spin representations) and vector bundles, 165, 166 Representation ring, 165, 190-192 character ring as, 168, 169 of compact group, 166-169 K(X) and KO(X) and, 166 as X-ring, 166, 173 real, 190-192 real Spin, 193-195 of SO(n), 187-190 of Sp(n), 182 of Spin(n), 187-190 of SU(n), 179 of a torus, 172-173 of U(n), 179 Riemannian metrics of vector bundle, 36 Ring, representation (see Representation ring) S-category, 205-206 Schur's lemma, 165 Semiring, ring completion, 103 Special orthogonal group (see Orthogonal group) Special unitary group (see Unitary group) Sphere(s), normal bundle to, 12 tangent bundle of, 12, 17,86-89,237 vector, fields on (see Vector fields, on spheres) Sphere bundles, 239-241 Spin group (Spin(n», 157-158 maximal tori, 183 real representations of, 193-195 representation ring of, 187-190 Weyl group of, 183 Spin representations, complex, 185-187, 195 and J(RPn), 224-225 real, 193-195 Index Splitting maps, 237 Stable equivalence (s-equivalence), 99, 104-105 Stability of classical groups, 82 Stiefel variety, 13, 75 as homogeneous space, 76-78 homotopy groups of, 83, 91 Stiefel-Whitney classes, axiomatic properties of, 234 definition, 234 by Steenrod squares, 243 dual,263 of a manifold, 261 mUltiplicative property of, 238, 239 relation of, to orientability, 246 Stiefel-Whitney numbers, 262 Subbundle, 11 Subspace, trivialization over, 110 Suspension, ,5, 287 decomposition, 292, 293 double, 291, 306 fibre bundles over, 8,5 sequences, 296, 309 Symmetric bilinear form, 142 Symmetric functions, 176-178 Symplectic group (Sp(n)), 39, 7,5 examples, 78 homotopy groups of, 82, 92-9,5 infinite, 76 maximal tori of, 180 representation ring of, 182 Weyl group of, 180 Tangent bundle, of manifold, 2,50 of projective space, 13, 17, 237 of sphere, 12, 17, 86-89, 237 Tholn isomorphism, 244 Euler class and, 241, 244 Thorn space(s), 203-205, 244 fibre homotopy type and, 213 Topological group, 39 Topology, compact-open, Torus (tori), maximal (see Maximal tori) representation ring of a, 172-173 Total space, 11 Transformation group, 39 Transition functions, 61 Trivialization over a subspace, 110 327 Unitary group U(n) and SU(n), 39, 7,5 examples, 78 homotopy groups of, 82, 92-9,5 infinite, 76 maximal tori of, 178 representation ring, 179 Weyl group of, 178 Universal bundle, ,52 for classical groups, 83 Milnor's construction of, ,52-.54 verification of universal property, ,5,5-,57 of vector bundles, 33, 84 Vector bundle(s), 23 atlas of charts for, 23, 60 classification, 33, 84 Euler characteristic of stable, 12,5 finite type, 31 homotopy classification of, 27-33, 101 induced, 26 isomorphism of, 2,5 metrics (riemannian and hermitian), 36 morphism of, 2.5 orientation of, 244 representations and, 16,5, 166 universal bundle of, 33, 84 Whitney sum of, 26 Vector fields, and Euler characteristic, 258 on spheres, 22, 139, 140, 1,56, 22,5 and coreducibility, 218-219 and J(RPk), 219-221 reducibility, 216-218 Weyl group, of compact group, 172 of SO(n), 182 of Sp(n), 180 of Spin (n), 183 of SU(n), 178 of U(n), 178 Whitney sum of vector bundles, 26 Wu's formula, 261 Y oneda representation theorem, 266 ... K-theory 11 2 Products in K-theory 11 6 The cofunctor L(X, A) 11 7 The difference morphism 11 9 Products in L(X, A) 12 1 The clutching construction 12 2 The cofunctor L,.(X, A) 12 4 Half-exact cofunctors 12 6... Maximal tori in Sp(n) 18 0 Formal identities in polynomial rings 18 0 The representation ring of Sp(n) 18 1 17 9 17 6 Contents 10 11 12 13 Maximal tori and the Weyl group of SO(n) 18 2 Maximal tori and... group Spin(k) 15 7 Exercises 15 8 12 THE ADAMS OPERATIONSi AND REPRESENTATIONS 10 11 15 9 X-rings 15 9 The Adams ,y-operations in X-ring 16 0 The -y' operations 16 2 Generalities on G-modules 16 3 The representation