Table Of Content82
Graduate Texts in Mathematics
Editorial Board
S. Axler F.W. Gehring P.R. Halmos
Springer Science+Business Media, LLC
Graduate Texts in Mathematics
T AKEUTUZARING. Introduction to 33 HIRSCH. Differential Topology.
Axiomatic Set Theory. 2nd ed. 34 SPITZER. Principles of Random Walk.
2 OXTOBY. Measure and Category. 2nd ed. 2nd ed.
3 SCHAEFER. Topological Vector Spaces. 35 WERMER. Banach Algebras and Several
4 HILTON/STAMMBACH. A Course in Complex Variables. 2nd ed.
Homological Algebra. 2nd ed. 36 KELLEY/NAMIOKA et al. Linear
5 MAC LANE. Categories for the Working Topological Spaces.
Mathematician. 37 MONK. Mathematical Logic.
6 HUGHES/PIPER. Projective Planes. 38 GRAUERT/FRlTZSCHE. Several Complex
7 SERRE. A Course in Arithmetic. Variables.
8 TAKEUTUZARING. Axiomatic Set Theory. 39 ARVESON. An Invitation to C*-Algebras.
9 HUMPHREYS. Introduction to Lie Algebras 40 KEMENy/SNELlJKNAPP. Denumerable
and Representation Theory. Markov Chains. 2nd ed.
10 COHEN. A Course in Simple Homotopy 41 APOSTOL. Modular Functions and
Theory. Dirichlet Series in Number Theory.
11 CONWAY. Functions of One Complex 2nd ed.
Variable I. 2nd ed. 42 SERRE. Linear Representations of Finite
12 BEALS. Advanced Mathematical Analysis. Groups.
\3 ANDERSON/FuLLER. Rings and Categories 43 GILLMAN/JERISON. Rings of Continuous
of Modules. 2nd ed. Functions.
14 GOLUBITSKy/GUILLEMIN. Stable Mappings 44 KENDIG. Elementary Algebraic Geometry.
and Their Singularities. 45 LOEVE. Probability Theory I. 4th ed.
15 BERBERIAN. Lectures in Functional 46 LOEVE. Probability Theory II. 4th ed.
Analysis and Operator Theory. 47 MOISE. Geometric Topology in
16 WINTER. The Structure of Fields. Dimensions 2 and 3.
17 ROSENBLATT. Random Processes. 2nd ed. 48 SACHS/WU. General Relativity for
18 HALMOS. Measure Theory. Mathematicians.
19 HALMOS. A Hilbert Space Problem Book. 49 GRUENBERG/WEIR. Linear Geometry.
2nd ed. 2nd ed.
20 HUSEMOLLER. Fibre Bundles. 3rd ed. 50 EDWARDS. Fermat's Last Theorem.
21 HUMPHREYS. Linear Algebraic Groups. 51 KLINGENBERG. A Course in Differential
22 BARNES/MACK. An Algebraic Introduction Geometry.
to Mathematical Logic. 52 HARTSHORNE. Algebraic Geometry.
23 GREUB. Linear Algebra. 4th ed. 53 MANIN. A Course in Mathematical Logic.
24 HOLMES. Geometric Functional Analysis 54 GRA VERlW ATKINS. Combinatorics with
and Its Applications. Emphasis on the Theory of Graphs.
25 HEWITT/STROMBERG. Real and Abstract 55 BROWN/PEARCY. Introduction to Operator
Analysis. Theory I: Elements of Functional
26 MANES. Algebraic Theories. Analysis.
27 KELLEY. General Topology. 56 MASSEY. Algebraic Topology: An
28 ZARISKUSAMUEL. Commutative Algebra. Introduction.
VoU. 57 CRowELlJFox. Introduction to Knot
29 ZARISKUSAMUEL. Commutative Algebra. Theory.
Vol.lI. 58 KOBLl1Z. p-adic Numbers, p-adic
30 JACOBSON. Lectures in Abstract Algebra I. Analysis, and Zeta-Functions. 2nd ed.
Basic Concepts. 59 LANG. Cyclotomic Fields.
31 JACOBSON. Lectures in Abstract Algebra 60 ARNOLD. Mathematical Methods in
II. Linear Algebra. Classical Mechanics. 2nd ed.
32 JACOBSON. Lectures in Abstract Algebra
III. Theory of Fields and Galois Theory. continued after index
Raoul Bott Loring W. Tu
Differential Forms in
Algebraic Topology
With 92 Illustrations
Springer
Raoul Bott Loring W. Tu
Mathematics Department Department of Mathematics
Harvard University Tufts University
Cambridge, MA 02138-2901 Medford, MA 02155-7049
USA USA
Editorial Board
S. Axler F.w. Gehring P.R. Halmos
Department of Department of Department of
Mathematics Mathematics Mathematics
Michigan State University University of Michigan Santa Clara University
East Lansing, MI 48824 Ann Arbor, MI 48109 Santa Clara, CA
95053
USA USA USA
Mathematics Subject Classifications (1991): 57Rxx, 58Axx, 14F40
Library of Congress Cataloging-in-Publication Data
Bott, Raoul, 1924-
Differential forms in algebraic topology
(Graduate texts in mathematics: 82)
Bibliography: p.
Includes index.
1. Differential topology. 2. Algebraic
topology. 3. 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 hereaf
ter 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.
9 8 7 6 5 4
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. Accord
ingly, 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 with
out 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 con
stant 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 com
pact 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
8-11
G
13-16 20-22
17 23
18
19
ix
Contents
Introduction
CHAPTER I
De Rham Theory
13
§1 The de Rham Complex on IR" 13
The de Rham complex 13
Compact supports 17
§2 The Mayer-Vietoris Sequence 19
The functor 0* 19
The Mayer-Vietoris sequence 22
The functor 0: and the Mayer-Vietoris sequence for compact supports 25
§3 Orientation and Integration 27
Orientation and the integral of a differential form 27
Stokes' theorem 31
§4 Poincare Lemmas 33
The Poincare lemma for de Rham cohomology 33
The Poincare lemma for compactly supported cohomology 37
The degree of a proper map 40
§5 The Mayer-Vietoris Argument 42
Existence of a good cover 42
Finite dimensionality of de Rham cohomology 43
Poincare duality on an orientable manifold 44
xi
xii Contents
The Kiinneth formula and the Leray-Hirsch theorem 47
The Poincare dual of a closed oriented submanifold 50
§6 The Thorn Isomorphism 53
Vector bundles and the reduction of structure groups 53
Operations on vector bundles 56
Compact cohomology of a vector bundle 59
Compact vertical cohomology and integration along the fiber 61
Poincare duality and the Thorn class 65
The global angular form, the Euler class, and the Thorn class 70
Relative de Rham theory 78
§7 The Nonorientable Case 79
The twisted de Rham complex 79
Integration of densities, Poincare duality, and the Thorn isomorphism 85
CHAPTER II
The Cech-de Rham Complex 89
§8 The Generalized Mayer-Vietoris Principle 89
Reformulation of the Mayer-Vietoris sequence 89
Generalization to countably many open sets and applications 92
§9 More Examples and Applications of the Mayer-Vietoris Principle 99
Examples: computing the de Rham cohomology from the
combinatorics of a good cover 100
Explicit isomorphisms between the double complex and de Rham and Cech 102
The tic-tac-toe proof of the Kiinneth formula 105
§10 Presheaves and Cech Cohomology 108
Presheaves 108
Cech cohomology 110
§11 Sphere Bundles 113
Orientability 114
The Euler class of an oriented sphere bundle 116
The global angular form 121
Euler number and the isolated singularities of a section 122
Euler characteristic and the Hopf index theorem 126
§12 The Thorn Isomorphism and Poincare Duality Revisited 129
The Thorn isomorphism 130
Euler class and the zero locus of a section 133
A tic-tac-toe lemma 135
Poincare duality 139
