Graduate Texts in Mathematics 20 Editorial Board J.H Ewing F.W Gehring P.R Halmos Graduate Texts in Mathematics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 TAKEUTJ/ZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFFER 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 TAKEUTJ/ZARING Axiometic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FULLER Rings and Categories of Modules 2nd ed GOLUBITSKY/GUILEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATI 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 HEWI'IT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKI/SAMUEL Commutative Algebra Vol I ZARISKI/SAMUEL Commutative Algebra Vol II JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory of Fields and Galois Theory 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/FRITZSCHE Several Complex Variables ARVESON An Invitation to C*-Aigebras 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 Lo~vE Probability Theory I 4th ed LoE:vE Probability Theory II 4th ed MoiSE Geometric Topology in Dimensions and continued after Index Dale H usemoller Fibre Bundles Third Edition Springer Science+Business Media, LLC Dale Husemoller Department of Mathematics Haverford College Haverford, PA 19041 USA Editorial Board H.Ewing Department of Mathematics Indiana University Bloomington, IN 47405 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 With four figures Mathematics Subject Classification (1991): 14F05, 14F15, 18F15, 18F25, 55RXX Library ofCongress Cataloging-in-Publieation Data Husemoller, Dale Fibre bundles/Dale Husemoller.-3rd ed p em.-(Graduate texts in mathematies;20) Inc1udes bibliographieal referenees and index ISBN 978-1-4757-2263-5 ISBN 978-1-4757-2261-1 (eBook) DOI 10.1007/978-1-4757-2261-1 Fiber bundles (Mathematies) Title II Series QA612.6.H87 1993 514'.224-de20 93-4694 Printed on acid-free paper First edition published by MeGraw-Hill, Ine., © 1966 by Dale Husemoller © 1994 Springer Science+Business Media New York Originally published by Springer-VerIag New York, Ine in 1994 Softcover reprint of the hardcover 3rd edition 1994 AII rights reserved This work may not be translated or eopied in whole or in part without the written permission of the publisher, Springer Scienee+Business Media, LLC, except for brief excerpts in conneetion with reviews or scholarly analysis Use in eonnection 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 Henry Krell; manufacturing supervised by Jacqui Ashri Typeset by Aseo Trade Typesetting Ltd., Hong Kong 98765432 ISBN 978-1-4757-2263-5 To the memory of my mother and my father Preface to the Third Edition In this edition, we have added two new chapters, Chapter on the gauge group of a principal bundle and Chapter 19 on the definition of Chern classes by differential forms These subjects have taken on special importance when we consider new applications of the fibre bundle theory especially to mathematical physics For these two chapters, the author profited from discussions with Professor M.S Narasimhan The idea of using the term bundle for what is just a map, but is eventually a fibre bundle projection, is due to Grothendieck The bibliography has been enlarged and updated For example, in the Seifert reference [1932] we find one of the first explicit references to the concept of fibrings The first edition of the Fibre Bundles was translated into Russian under the title "PaccnoeHHhie llpocTpaHCTBa" in 1970 by B A McKOBCKHX with general editor M M llocTHHKOBa The remarks and additions of the editor have been very useful in this edition of the book The author is very grateful to A Voronov, who helped with translations of the additions from the Russian text Part of this revision was made while the author was a guest of the Max Planck Institut from 1988 to 89, the ETH during the summers of 1990 and 1991, the University of Heidelberg during the summer of 1992, and the Tata Institute for Fundamental Research during January 1990, 1991, and 1992 It is a pleasure to acknowledge all these institutions as well as the Haverford College Faculty Research Fund 1993 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 profitted 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 1974 Dale Husemoller Preface to the First Edition 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 inK-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 the xii Preface to the First Edition 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 1966 Dale Husemoller Bibliography 341 La transgression dans les espaces fibres principaux, Compt Rend., 232: 2392-2394 (1951) Sur Ia cohomologie des espaces fibres principaux et des espaces homogenes de groups de Lie compacts, Ann Math., 51: 115-207 (1953) Topology of Lie groups and characteristic classes, 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309-313 (1952) Moore, J C.: Seminaire Henri Cartan, 1954/55, expose 22 The double suspension and p-primary components of the homotopy groups of spheres, Boletin de Ia Sociedad Matematica Mexicana, 1956, vol 1, pp 28-37 Morin, B.: Champs de vecteurs sur les spheres d'apres J F Adams, Seminaire Bourbaki, no 233, 1961-1962 Narasimhan, M.S., and S Ramanan: Existence of universal connections, American Journal of Mathematics, 83:563-572 (1961) Existence of universal connections II, American Journal of Mathematics, 85: 223231 (1963) Narasimhan, M S., and C S Seshadri: Stable and unitary vector bundles on a compact Riemann surface, Ann of Math., 82: 540-567 (1965) Pontrjagin, L.: "Topological Groups," Princeton University Press, Princeton, N.J., 1939 Characteristic cycles on differentiable manifolds, Mat Sb N.S., (63), 21: 233-284 (1947) Bibliography 345 Puppe, D.: l Homotopiemengen und ihre induzierten Abbildungen I Math Z., 69: 299-344 (1958) Faserraume, University of Saarland, Saarbrucken, Germany, 1964 Quillen, D.: l Some remarks on etale homotopy theory and a conjecture of Adams, Topology, (1968), 111-116 The Adams conjecture, Topology, 10 (1971), 67-80 Cohomology of groups, Acts, Congres Intern Math 1970, t.2: 47-51 Higher algebraic K-theory I, Springer Lecture Notes in Math 341:85-147 (1973) Superconnection character forms and the Cayley transform, Topology, 21:211-238 (1988) Algebra cochains and cyclic cohomology, Pub Math de IHES, 68139-174 (1990) Sanderson, B J.: l Immersions and embeddings of projective spaces, Proc London Math Soc., (3)14: 137-153 (1964) A non-immersion theorem for real projective space, Topology, 2: 209-211 (1963) Segal, G.: l Configuration spaces and iterated loop spaces, Invent Math 21: 213-221 (1973) Categories and cohomology theories, Topology, 13: 293-312 (1974) Seifert, H.: l Topologie 3-dimensionaler gefaserter Raume, Acta Math., 60: 147-238 (1932) Serre, J P.: l Homologie singuliere des espaces fibres, Ann Math., 54:425-505 (1951) Groupes d'homotopie et classes de groupes abeliens, Ann Math., 58: 258-294 (1953) Cohomologie, modulo des complexes d'Eilenberg-MacLane, Comment Math Helv., 27: 198-232 (1953) Shih, W.: l Appendix II, Seminar on the Atiyah-Singer index theorem by R S Palais, Annals of Mathematics Studies, Number 57 Spanier, E.: l A formula of Atiyah and Hirzebruch, Math Z., 80: 154-162 (1962) Function spaces and duality, Ann Math., 70: 338-378 (1959) "Algebraic Topology," McGraw-Hill Book Company, New York, 1966 Spanier, E., and J H C Whitehead: l Duality in homotopy theory, Mathematika, 2: 56-80 (1955) Stasheff, J.: l A classification theorem for fibre spaces, Topology, 2: 239-246 (1963) Steenrod, N E.: l Classification of sphere bundles, Ann Math., 45: 294-311 (1944) "Topology of Fibre Bundles," Princeton Mathematical Series, Princeton University Press, Princeton, N.J., 1951 Cohomology operations, Ann Math Studies 50, 1962 Steenrod, N E., and J H C Whitehead: l Vector fields on then-sphere, Proc Nat/ Acad Sci U.S.A., 37: 58-63 (1951) 346 Bibliography Stiefel, E.: Richtungsfelder und Fernparakkekusnus in n-dimensionalen Mannigfaltigkeiten, Comment Math Helv., 8: 3-51 (1936) Thorn, R.: Espaces fibres en spheres et carres de Steenrod, Ann Ecole Norm Sup., 69: 109-182 (1952) Quelques proprietes globales des varietes differentiables, Comment Math Helv., 28: 17-86 (1954) Thomason, R.: Algebraic K-theory and etale cohomology, Ann Scient Ec Norm Sup 13:437-552 (1980) Toda, H.: Generalized Whitehead products and homotopy groups of spheres, J Jnst Polytech., Osaka City Univ., 3: 43-82 (1952) On the double suspension E , J-Inst Polytech., Osaka City Univ., Ser A, 7: 103145 (1956) Reduced join and Whitehead product, J Inst Polytech., Osaka City Univ., Ser A, 8: 15-30 (1957) Waldhausen, F.: Algebraic K-theory of spaces, localization and the chromatic filtration of stable homotopy, SLN 1051: 173-195 (1984) Whitehead, G W.: On the homotopy groups of spheres and rotation groups, Ann Mat h., 43: 634-640 (1942) Homotopy properties of the real orthogonal groups, Ann Math., (2)43: 132-146 (1942) On families of continuous vector fields over spheres, Ann Math., (2)47: 779-785 (1946) A generalization of the Hopfinvariant, Ann Math., 51: 192-238 (1950) Generalized homology theories, Trans Am Math Soc., 102: 227-283 (1962) Whitehead, J H C.: On the groups n:,(V.,m) and sphere bundles, Proc London Math Soc., (2)48: 243291 (1944) Combinatorial homotopy: I, Bull Am Math Soc., 55: 213-245 (1949) Combinatorial homotopy: II, Bull Am Math Soc~ 55:453-496 (1949) Whitney, H.: Sphere spaces, Proc Nat/ Acad Sci U.S.A., 21:462-468 (1935) Topological properties of differentiable manifolds, Bull Am Math Soc., 43: 785805 (1937) On the theory of sphere bundles, Proc Nat/ Acad Sci U.S.A., 26:148-153 (1940) On the topology of differentiable manifolds, "Lectures in Topology," The University of Michigan Press, Ann Arbor, Mich., 1941 Wu, Wen-Tsun: On the product of sphere bundles and the duality theorem modulo two, Ann Math., 49:641-653 (1948) Sur Ies classes caracteristiques d'un espace fibrees en spheres, Compt Rend., 227: 582-584 (1948) Les i-carres dans une variete grassmannienne, Compt Rend., 230: 918-920 (1950) Classes caracteristiques et i-carres d'une variete, Compt Rend., 230: 508-511 (1950) Bibliography 347 Sur les classes caracteristiques des structures fibries spheriques, Actualities Sci Inc., no 1183-Publ Inst Math Univ Strasbourg 11, pp 5-98, 155-156, Hermann & Cie, Paris, 1952 Yokota, 1.: On the cellular decompositions of unitary groups, J Inst Polytech., Osaka City Univ., 7: 30-49 (1956) Index Adams conjecture, 240 Adams operations, 172 in A.-rings, 172-174 on real spin representations, 207, 238 Ale.xander duality theorem, 271 Atiyah duality theorem, 221, 222 Atlas of charts, for fibre bundle, 62 for manifold, 263 Automorphisms of principal bundles, 79-81 Base space, 11 Bianchi identity, 287 Bilinear form, symmetric, 154 Bott periodicity, 140, 149 and integrality theorem, 150, 236, 307, 308 Bundle(s), 11 collapsed, 122 fibre [see Fibre bundle(s)] fibre product of, 16 G-, 42 Hopf, 141 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-, 11 tangent (see Tangent bundle) Bundle(s), universal (see Universal bundle) vector [see Vector bundle(s)] Character ring, 180 representation ring as, 180, 181 Characteristic classes, 295, 296, 300, 304 calculations, on canonical line bundles, 248, 249, 259-261 on tangent bundles of S", RP", and CP", 250, 251, 306, 307 Chern (see Chern classes) complex, 297-299 in dimension n, 296, 298 Euler (see Euler class) Pontrjagin (see Pontrjagin classes) real, mod 2, 300-301 2-divisible, 301-304 and representations, 309-311 Stiefel-Whitney (see Stiefel-Whitney classes) Characteristic map, 101-104 Index Charts, of fibre bundles, 62 of manifolds, 263 of vector bundles, 24, 62 Chern classes, 249, 296-299 axiomatic properties of, 249 definition, 249 multiplicative property of, 252 Chern forms, 286 curvature, 284,285 definition, 284 homotopy formula, 288-291 Chern-Simons invariants, 290, 291 Classical groups, 87-90 classifying spaces for, 95, 96 examples, 89, 90 homotopy groups of, 95, 104-107 infinite, 88 stability of, 94, 95 universal bundle for, 95, 96 Classifying spaces, 63 for classical groups, 95, 96 cohomology of, 297, 300, 302 of reduced structure group, 77 of vector bundles, 31, 32, 96 Clifford algebras, 156-161 calculations of, 158 table, 161 Clifford modules, 161-163 table, 163 tensor products of, 166-168 Clutching construction, 134-136 Clutching maps, 135 Laurent, 142 linear, 145-148 polynomial, 143-145 Cobordism, 276-278 Cofunctor, half-exact (see Half-exact cofunctor) coH-space, Cokernel of morphism, 35-37 Collapsed bundle, 122 Compact group, 179 maximal tori of, 182-184 rank of, 183 representation ring of, 179-181 Weyl group of, 184 Compact-open topology, Cone over a space, Connections on a vector bundle, 349 Coreducible spaces, 228 relation to vector fields, 179-181 Covariant derivative, 292 Cross section(s), 12 and Euler class, 257 of fibre bundles, 48 prolongation of, 21 CW-complex(es), homotopy classification over, 58 de Rham cohomology, 281 Difference isomorphism 131-133 Differential forms, 280-283 Duality, in manifolds, 269-272 Duality theorem, Poincare, 271 Eilenberg-MacLane spaces, 83-86 Euclidean inner product, 12 Euclidean norm, 12 Euclidean space, orientation in, 266-267 Euler characteristic, definition, 274-275 and Euler class, 274-275 of stable vector bundles, 137 and vector fields, 275 Euler class, 254, 301-303 and cross sections, 257 definition, 254 and Euler characteristic, 274, 275 of a manifold, 274, 275 multiplicative property, 256, 257 and Thorn isomorphism, 258 Fibre, 11 of a fibre bundle, 45 of a principal bundle, 43 Fibre bundle(s), 45 atlas of charts for, 62 automorphisms, 61, 79-81 classification of, 56-59 cross section of, 48 locally trivial, 47 morphism of, 46 over suspension, 85 trivial, 47 Fibre homotopy equivalence, 223, 224, 312,313 350 Fibre homotopy type, 223,224 312,317 stable, 223-228 and Thorn spaces, 227, 228 Fibre maps, 7, 312-313 Fibre product, 16 Gauge group classifying space of, 83 calculation of, 81 definiton, 79-81 universal bundle of, 83 Gauss map, 33 G-module, 176 direct sum of, 176 exterior product of, 176 morphism of, 176 semisimple, 177, 179 tensor product of, 176 Grassman manifold (or variety), 13, 25, 34 cohomology of, 297, 300, 302 as homogeneous space, 90, 91 Group(s), classical (see Classical groups) compact (see Compact group) homotopy (see Homotopy groups) Group(s), linear, 40 Group(s), orthogonal (see Orthogonal group) reduction of structure, 77 spin (Spin(n)) (see Spin group) symplectic (see Symplectic group) topological, 40 transformation, 40 unitary (see Unitary group) G-space, 40 morphism of, 41 principal, 42 Gysin sequence, 255 Half-exact cofunctor, 138 Puppe sequence of, 139 Hermitian metrics of vector bundle, 37 Homotopy classification, over CW-complexes, 58 of principal bundles, 56-58 of vector bundle, 33-35, 113, ll4 Index Homotopy equivalence, fibre,223,224,285, 312,313 Homotopy formula for: connections, 288-291 differential forms, 283 Homotopy groups, of classical groups, 94, 104-107 of O(n), 94, 104-107 of SO(n), 94, 104-107 of Sp(n), 94, 104-107 of Stiefel variety, 95, 103, 104 of SU(n), 94, 104-107 of U(n), 94, 104-107 Homotopy type, fibre (see Fibre homotopy type) Hopf bundle, 142, 143 Hopfinvariant, 210-216, 309, 326-328 H-space, Integrality theorem of Bott, 307, 308 J(X), 224 calculation of J(RP"), 237-239 of J(Sk), 225-227 K-cup product, 128 k-space, K(X) and KO(X), 114-117, 120 Bott periodicity of, 140, 148-150, 236 as A.-ring, 171, 186-188 and representation ring, 180-181 as a ring, 128 K(X) and io(X), ll4-ll7 calculation of K(S"), 187, 188 corepresentation of, ll8, 119 table of results for KO(S") and KO(RP"), 235-237 Kiinneth formula, 85, 86 l-ring(s), 171 Adams operations in, 172 K(X) and KO(X) as, 171, 186-188 y-operations in, 175 representation ring as, 180, 181, 185 split, 174 351 Index Leray-Hirsch theorem, 245 Levi-Civita, connection, 291-293 Linear groups, 40 Loop space, Normal bundle, of an immersion, 278 of a map, 265 to the sphere, 13 Numerable covering, 49, 312 Manifold(s), 262-279 atlas of charts for, 262, 263 duality in, 269-272 Euler class of, 274 fundamental class of, 268 Grassman [see Grassman manifold (or variety)] orientation of, 267-269 Stiefel-Whitney classes of, 275, 276 tangent bundle to, 264 Thorn class of, 272-274 Map(s), clutching (see Clutching maps) fibre, 7, 312, 313 Gauss, 33 normal bundle of, 265 splitting, 251 Map space, Mapping cone, 125 Mapping cylinder, 125 Maximal tori, of compact groups, 182-184 of SO(n), 195, 196 of Sp(n), 193 of Spin(n), 196, 197 of SU(n), 191 of U(n), 191 Mayer-Vietoris sequence, 246, 268270 Milnor's construction of a universal bundle, 54-56 verification of universal property, 56-58 Morphism, B-, 14 of bundles, 14 cokernel of, 36 of fibre bundles, 45 of G-module, 176 of G-space, 42 image of, 36 kernel of, 36 local representation of, 66 of principal bundles, 43 of vector bundles, 26 Orientation, in euclidean space, 266, 267 of manifolds, 267-269 of vector bundles, 260 Orientation class, 267, 268 Orthogonal group (O(n)) and (SO(n)), 40,87 examples, 90 homotopy groups of, 94, 104-107 infinite, 88 maximal tori of, 195, 196 representation ring of, 200-203 Weyl group of, 195, 196 Orthogonal multiplication, 152, 153 Orthogonal splitting, 155, 156 Path space, Periodicity, Bott (see Bott periodicity) Poincare duality theorem, 271 Pontrjagin classes, 259, 260, 301-304 Principal bundle(s), 42 homotopy classification of, 56-58 induced, 43 morphism of, 42 numerable, 49 Products, of bundles, 15-17 euclidean inner, 13 of G-module, exterior, 176 tensor, 176 reduced, Projection, 11 Projective space, tangent bundle of, 14, 17, 251 Puppe sequence, 125-127, 139, 220, 221 Quadratic form, 154 Rank of a compact group, 183 Reduced product, Reducible spaces, 227 relation to vector fields, 230-232 352 Representation(s), 176-178 and characteristic classes, 309-311 local, of morphism, 66 real, of Spin(n), 203-205 semisimple, 179, 180 spin (see Spin representations) and vector bundles, 177-178 Representation ring, 177-178, 203-205 character ring as, 180, 181 of compact group, 179-181 K(X) and KO(X) and, 177, 178 as A.-ring, 177, 178, 186, 187 real, 203-206 real-Spin, 203-205 of SO(n), 200-203 of Sp(n), 195 of Spin(n), 200-203 of SU(n), 192 of a torus, 185, 186 of U(n), 192 Riemann connection, 291-293 Riemannian metrics of vector bundle, 37,38 Ring, representation (see Representation ring) S-category, 219-220 Schur's lemma, 179 Semiring, ring completion, 115, 116 Special orthogonal group (see Orthogonal group) Special unitary group (see Unitary group) Sphere(s), normal bundle to, 13 tangent bundle of, 13, 17,98-100,251 vector, fields on (see Vector fields, on spheres) Sphere bundles, 253-255 Spin group (Spin(n)), 169, 170 maximal tori, 196, 197 real representations of, 203-205 representation ring of, 200-203 Weyl group of, 196, 197 Spin representations, complex, 198-200, 206-208 and J(RP"), 234-239 real, 206-208 Splitting maps, 251 Index Stable equivalence (s-equivalence), 111, 118-120 Stability of classical groups, 94 Stiefel variety, 13, 91, 92 as homogeneous space, 87-89 homotopy groups of, 91-93, 95 Stiefel-Whitney classes, axiomatic properties of, 24 definition, 247 by Steenrod squares, 257 dual, 278 of a manifold, 275 multiplicative property of, 252, 253 relation of, to orientability, 278 Stiefel-Whitney numbers, 276 Subbundle, 11 Subspace, trivialization over, 122 Suspension, 5, 314 decomposition, 319-321 double, 318, 333-337 fibre bundles over, 97 sequences, 322-323 Symmetric bilinear form, 154 Symmetric functions, 189-191,285, 286 Symplectic group (Sp(n)), 40, 87 examples, 90 homotopy groups of, 94, 104-107 infinite, 88 maximal tori of, 193 representation ring of, 195 Weyl group of, 193 Tangent bundle, of manifold, 263 of projective space, 14, 17,251 of sphere, 13, 17, 98-100, 251 Thorn isomorphism, 258 Euler class and, 255, 258 Thorn space(s), 217-219, 258 fibre homotopy type and, 227 Topological group, 40 Topology, compact-open, Torus (tori), maximal (see Maximal tori) representation ring of a, 185, 186 Total space, 11 Transformation group, 40 Transition functions, 62, 63 Trivialization over a subspace, 122 353 Index Unitary group U(n) and SU(n), 40, 87 examples, 90 homotopy groups of, 94, 104-107 infinite, 88 maximal tori of, 191 representation ring, 192 Weyl group of, 191 Universal bundle, 54, 55 for classical groups, 95 Milnor's construction of, 54-56 verification of universal property, 56-58 of vector bundles, 35, 96 Vector bundle(s), 24 atlas of charts for, 24, 62, 63 classification, 34, 35, 96 Euler characteristic of stable, 136-137 finite type, 32 homotopy classification of, 28-35, 113 induced, 27, 28 isomorphism of, 26 metrics (riemannian and hermitian), 37-38 morphism of, 26 orientation of, 244, 285, 286 representations and, 177, 178 universal bundle of, 34, 35, 97 Whitney sum of, 27 Vector fields, and Euler characteristic, 274,275 on spheres, 24, 151, 152, 168, 239 and coreducibility, 232-233 and J(RPk), 233-235 reducibility, 230-232 Wei!, Andre, 280 Weyl group, of compact group, 184 of SO(n), 195, 196 of Sp(n), 193 of Spin(n), 196, 197 of SU(n), 178, 191 of U(n), 178, 191 Whitney sum of vector bundles, 27 Wu's formula, 275 Yoneda representation theorem, 294, 295 Graduate Texts in Mathematics continued from page II 48 49 50 51 52 53 54 55 56 57 58 59 60 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 92 93 94 95 96 SACHS/Wu General Relativity for Mathematicians GRUENBERG/WEIR Linear Geometry 2nd ed EDWARDS Fermat's Last Theorem KLINGENBERG A Course in Differential Geometry HARTSHORNE Algebraic Geometry 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Geometric Topology in Dimensions and continued after Index Dale H usemoller Fibre Bundles Third Edition Springer Science+Business Media, LLC Dale Husemoller Department of Mathematics Haverford College... Bundles Definition of Fibre Bundles Functorial Properties of Fibre Bundles Trivial and Locally Trivial Fibre Bundles Description of Cross Sections of a Fibre Bundle Numerable