Graduate Texts in Mathematics 82 Editorial Board S Axler F.W Gehring P.R Halmos Springer Science+Business Media, LLC Graduate Texts in Mathematics 10 11 12 \3 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 TAKEUTUZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable I 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 HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKUSAMUEL Commutative Algebra VoU ZARISKUSAMUEL Commutative Algebra Vol.lI 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 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/SNELlJKNAPP 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 I 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 GRA VERlWATKINS 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 CRowELlJFox Introduction to Knot Theory 58 KOBLl1Z 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 Raoul Bott Loring W Tu Differential Forms in Algebraic Topology With 92 Illustrations Springer Loring W Tu Department of Mathematics Tufts University Medford, MA 02155-7049 USA Raoul Bott Mathematics Department Harvard University Cambridge, MA 02138-2901 USA Editorial Board S Axler Department of Mathematics Michigan State University East Lansing, MI 48824 95053 USA F.w Gehring Department of Mathematics University of Michigan Ann Arbor, MI 48109 P.R Halmos Department of Mathematics Santa Clara University Santa Clara, CA USA USA Mathematics Subject Classifications (1991): 57Rxx, 58Axx, 14F40 Library of Congress Cataloging-in-Publication Data Bott, Raoul, 1924Differential forms in algebraic topology (Graduate texts in mathematics: 82) Bibliography: p Includes index Differential topology Algebraic topology Differential forms I Tu Loring W II Title III Series QA613.6.B67 514'.72 81-9172 Printed on acid-free paper © 1982 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc in 1982 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC , except for brief excerpts in connection with reviews or scholarly analysis 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 Production managed by Bill Imbornoni; manufacturing supervised by Jacqui Ashri ISBN 978-1-4419-2815-3 ISBN 978-1-4757-3951-0 (eBook) DOI 10.1007/978-1-4757-3951-0 SPIN 10635035 For Phyllis Bott and Lichu and Tsuchih Tu Preface The guiding principle in this book is to use differential forms as an aid in exploring some of the less digestible aspects of algebraic topology Accordingly, we move primarily in the realm of smooth manifolds and use the de Rham theory as a prototype of all of cohomology For applications to homotopy theory we also discuss by way of analogy cohomology with arbitrary coefficients Although we have in mind an audience with prior exposure to algebraic or differential topology, for the most part a good knowledge of linear algebra, advanced calculus, and point-set topology should suffice Some acquaintance with manifolds, simplicial complexes, singular homology and cohomology, and homotopy groups is helpful, but not really necessary Within the text itself we have stated with care the more advanced results that are needed, so that a mathematically mature reader who accepts these background materials on faith should be able to read the entire book with the minimal prerequisites There are more materials here than can be reasonably covered in a one-semester course Certain sections may be omitted at first reading without loss of continuity We have indicated these in the schematic diagram that follows This book is not intended to be foundational; rather, it is only meant to open some of the doors to the formidable edifice of modern algebraic topology We offer it in the hope that such an informal account of the subject at a semi-introductory level fills a gap in the literature It would be impossible to mention all the friends, colleagues, and students whose ideas have contributed to this book But the senior author would like on this occasion to express his deep gratitude, first of all to his primary topology teachers E Specker, N Steenrod, and vii viii Preface K Reidemeister of thirty years ago, and secondly to H Samelson, A Shapiro, I Singer, l-P Serre, F Hirzebruch, A Borel, J Milnor, M Atiyah, S.-s Chern, J Mather, P Baum, D Sullivan, A Haefliger, and Graeme Segal, who, mostly in collaboration, have continued this word of mouth education to the present; the junior author is indebted to Allen Hatcher for having initiated him into algebraic topology The reader will find their influence if not in all, then certainly in the more laudable aspects of this book We also owe thanks to the many other people who have helped with our project: to Ron Donagi, Zbig Fiedorowicz, Dan Freed, Nancy Hingston, and Deane Yang for their reading of various portions of the manuscript and for their critical comments, to Ruby Aguirre, Lu Ann Custer, Barbara Moody, and Caroline Underwood for typing services, and to the staff of Springer-Verlag for its patience, dedication, and skill F or the Revised Third Printing While keeping the text essentially the same as in previous printings, we have made numerous local changes throughout The more significant revisions concern the computation ofthe Euler class in Example 6.44.1 (pp 75-76), the proof of Proposition 7.5 (p 85), the treatment of constant and locally constant presheaves (p 109 and p 143), the proof of Proposition 11.2 (p 115), a local finite hypothesis on the generalized Mayer-Vietoris sequence for compact supports (p 139), transgressive elements (Prop 18.13, p 248), and the discussion of classifying spaces for vector bundles (pp 297-3(0) We would like to thank Robert Lyons, Jonathan Dorfman, Peter Law, Peter Landweber, and Michael Maltenfort, whose lists of corrections have been incorporated into the second and third printings RAOUL BOTT LORINOTu Interdependence of the Sections 1-6 D G 8-11 13-16 20-22 17 23 18 19 ix Contents Introduction CHAPTER I De Rham Theory §1 The de Rham Complex on IR" 13 The de Rham complex Compact supports 13 17 §2 The Mayer-Vietoris Sequence §3 19 The functor 0* The Mayer-Vietoris sequence The functor 0: and the Mayer-Vietoris sequence for compact supports 19 22 25 Orientation and Integration 27 27 Orientation and the integral of a differential form Stokes' theorem §4 13 31 Poincare Lemmas 33 The Poincare lemma for de Rham cohomology The Poincare lemma for compactly supported cohomology The degree of a proper map 33 §5 The Mayer-Vietoris Argument Existence of a good cover Finite dimensionality of de Rham cohomology Poincare duality on an orientable manifold 37 40 42 42 43 44 xi xii Contents The Kiinneth formula and the Leray-Hirsch theorem The Poincare dual of a closed oriented submanifold §6 The Thorn Isomorphism Vector bundles and the reduction of structure groups Operations on vector bundles Compact cohomology of a vector bundle Compact vertical cohomology and integration along the fiber Poincare duality and the Thorn class The global angular form, the Euler class, and the Thorn class Relative de Rham theory §7 The Nonorientable Case The twisted de Rham complex Integration of densities, Poincare duality, and the Thorn isomorphism 47 50 53 53 56 59 61 65 70 78 79 79 85 CHAPTER II The Cech-de Rham Complex §8 The Generalized Mayer-Vietoris Principle 89 Reformulation of the Mayer-Vietoris sequence Generalization to countably many open sets and applications 89 92 §9 More Examples and Applications of the Mayer-Vietoris Principle §10 §11 §12 89 99 Examples: computing the de Rham cohomology from the combinatorics of a good cover Explicit isomorphisms between the double complex and de Rham and Cech The tic-tac-toe proof of the Kiinneth formula 100 102 105 Presheaves and Cech Cohomology 108 Presheaves Cech cohomology 108 110 Sphere Bundles 113 Orientability The Euler class of an oriented sphere bundle The global angular form Euler number and the isolated singularities of a section Euler characteristic and the Hopf index theorem 114 116 121 122 126 The Thorn Isomorphism and Poincare Duality Revisited 129 The Thorn isomorphism Euler class and the zero locus of a section A tic-tac-toe lemma Poincare duality 130 133 135 139 Index Abelian group structure of a finitely generated Abelian group Action (See also Action of 7T) on 7Tq ) adjoint action 301 effective action 48 Action of 7T) on 7Tq 211 for the orthogonal groups 302 Adjoint action 301 Algebra (See also Graded algebra) divided polynomial algebra 205 exterior algebra 205 free algebra 259 Alternating difference 110 Alternating sum formula 186 Angular form 70 global angular form 121, 122 Antiderivation 14, 174, 175 Antipodal map 75 Associated flag bundle 282 cohomology ring 284 Poincare series 285 Associated graded complex 156 Atlas 20 Attaching cells 217 CW-complex 219 homology property 219 homotopy property 217 Averaging 213,304 Back r-face 192 Barycenter 142 Barycentric subdivision 142 Base points dependence of homotopy groups on 210 Bidegrees in a cohomology spectral sequence 164 in a homology spectral sequence 197 Blow-up 268 Bott, Raoul 304,305 Boundary 30 induced map on 18 of a manifold 30 Bo~ndary map Cech boundary operator 186 for singular chains 184 in homotopy sequence 209, 254 Brown, Edgar 10 Bump form 25, 40, 68 Bundle map 54 Cartan, Henri Category 20 of commutative differential graded algebras 20 of differentiable manifolds 20, 59 of Euclidean spaces 20 of open sets 109 • of topological spaces 59, 182