Dieck_titelei 1.8.2008 13:23 Uhr Seite 1 Dieck_titelei 1.8.2008 13:23 Uhr Seite 2
EMS Textbooks in Mathematics
EMS Textbooks in Mathematics is a book series aimed at students or professional mathematici- ans seeking an introduction into a particular field. The individual volumes are intended to provide not only relevant techniques, results and their applications, but afford insight into the motivations and ideas behind the theory. Suitably designed exercises help to master the subject and prepare the reader for the study of more advanced and specialized literature.
Jørn Justesen and Tom Høholdt, A Course In Error-Correcting Codes Markus Stroppel, Locally Compact Groups Peter Kunkel and Volker Mehrmann, Differential-Algebraic Equations Dorothee D. Haroske and Hans Triebel, Distributions, Sobolev Spaces, Elliptic Equations Thomas Timmermann, An Invitation to Quantum Groups and Duality Marek Jarnicki and Peter Pflug, First Steps in Several Complex Variables: Reinhardt Domains Oleg Bogopolski, Introduction to Group Theory Dieck_titelei 1.8.2008 13:23 Uhr Seite 3
Tammo tom Dieck Algebraic Topology Dieck_titelei 1.8.2008 13:23 Uhr Seite 4
Author:
Tammo tom Dieck Mathematisches Institut Georg-August-Universität Göttingen Bunsenstrasse 3–5 37073 Göttingen Germany
E-mail: [email protected]
2000 Mathematics Subject Classification: 55-01, 57-01
Key words: Covering spaces, fibrations, cofibrations, homotopy groups, cell complexes, fibre bundles, vector bundles, classifying spaces, singular and axiomatic homology and cohomology, smooth manifolds, duality, characteristic classes, bordism.
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
ISBN 978-3-03719-048-7
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.
© 2008 European Mathematical Society
Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org
Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printed on acid-free paper produced from chlorine-free pulp. TCF °° Printed in Germany
9 8 7 6 5 4 3 2 1 Preface
Algebraic topology is the interplay between “continuous” and “discrete” mathe- matics. Continuous mathematics is formulated in its general form in the language of topological spaces and continuous maps. Discrete mathematics is used to express the concepts of algebra and combinatorics. In mathematical language: we use the real numbers to conceptualize continuous forms and we model these forms with the use of the integers. For example, our intuitive idea of time supposes a continuous process without gaps, an unceasing succession of moments. But in practice we use discrete models, machines or natural processes which we define to be periodic. Likewise we conceive of a space as a continuum but we model that space as a set of discrete forms. Thus the essence of time and space is of a topological nature but algebraic topology allows their realizations to be of an algebraic nature. Classical algebraic topology consists in the construction and use of functors from some category of topological spaces into an algebraic category, say of groups. But one can also postulate that global qualitative geometry is itself of an algebraic nature. Consequently there are two important view points from which one can study algebraic topology: homology and homotopy. Homology, invented by Henri Poincaré, is without doubt one of the most inge- nious and influential inventions in mathematics. The basic idea of homology is that we start with a geometric object (a space) which is given by combinatorial data (a simplicial complex). Then the linear algebra and boundary relations determined by these data are used to produce homology groups. In this book, the chapters on singular homology, homology, homological algebra and cellular homology constitute an introduction to homology theory (construction, axiomatic analysis, classical applications). The chapters require a parallel reading – this indicates the complexity of the material which does not have a simple intuitive explanation. If one knows or accepts some results about manifolds, one should read the construction of bordism homology. It appears in the final chapter but offers a simple explanation of the idea of homology. The second aspect of algebraic topology, homotopy theory, begins again with the construction of functors from topology to algebra. But this approach is important from another view point. Homotopy theory shows that the category of topological spaces has itself a kind of (hidden) algebraic structure. This becomes immediately clear in the introductory chapters on the fundamental group and covering space theory. The study of algebraic topology is often begun with these topics. The notions of fibration and cofibration, which are at first sight of a technical nature, are used to indicate that an arbitrary continuous map has something like a kernel and a cokernel – the beginning of the internal algebraic structure of topology. (The chapter on homotopy groups, which is essential to this book, should also be studied vi Preface for its applications beyond our present study.) In the ensuing chapter on duality the analogy to algebra becomes clearer: For a suitable class of spaces there exists a duality theory which resembles formally the duality between a vector space and its dual space. The first main theorem of algebraic topology is the Brouwer–Hopf degree the- orem. We prove this theorem by elementary methods from homotopy theory. It is a fairly direct consequence of the Blakers–Massey excision theorem for which we present the elementary proof of Dieter Puppe. Later we indicate proofs of the de- gree theorem based on homology and then on differential topology. It is absolutely essential to understand this theorem from these three view points. The theorem says that the set of self-maps of a positive dimensional sphere under the homotopy relation has the structure of a (homotopically defined) ring – and this ring is the ring of integers. The second part of the book develops further theoretical concepts (like coho- mology) and presents more advanced applications to manifolds, bundles, homotopy theory, characteristic classes and bordism theory. The reader is strongly urged to read the introduction to each of the chapters in order to obtain more coherent infor- mation about the contents of the book. Words in boldface italic are defined at the place where they appear even if there is no indication of a formal definition. In addition, there is a list of standard or global symbols. The problem sections contain exercises, examples, counter-examples and further results, and also sometimes ask the reader to extend concepts in further detail. It is not assumed that all of the problems will be completely worked out, but it is strongly recommended that they all be read. Also, the reader will find some familiarity with the full bibliography, not just the references cited in the text, to be crucial for further studies. More background material about spaces and manifolds may, at least for a while, be obtained from the author’s home page. I would like to thank Irene Zimmermann and Manfred Karbe for their help and effort in preparing the manuscript for publication.
Göttingen, September 2008 Tammo tom Dieck Contents
Preface v
1 Topological Spaces 1 1.1 Basic Notions ...... 1 1.2 Subspaces. Quotient Spaces ...... 5 1.3 Products and Sums ...... 8 1.4 Compact Spaces ...... 11 1.5 Proper Maps ...... 14 1.6 Paracompact Spaces ...... 15 1.7 Topological Groups ...... 15 1.8 Transformation Groups ...... 17 1.9 Projective Spaces. Grassmann Manifolds ...... 21
2 The Fundamental Group 24 2.1 The Notion of Homotopy ...... 25 2.2 Further Homotopy Notions ...... 30 2.3 Standard Spaces ...... 34 2.4 Mapping Spaces and Homotopy ...... 37 2.5 The Fundamental Groupoid ...... 41 2.6 The Theorem of Seifert and van Kampen ...... 45 2.7 The Fundamental Group of the Circle ...... 47 2.8 Examples ...... 52 2.9 Homotopy Groupoids ...... 58
3 Covering Spaces 62 3.1 Locally Trivial Maps. Covering Spaces ...... 62 3.2 Fibre Transport. Exact Sequence ...... 66 3.3 Classification of Coverings ...... 70 3.4 Connected Groupoids ...... 72 3.5 Existence of Liftings ...... 76 3.6 The Universal Covering ...... 78
4 Elementary Homotopy Theory 81 4.1 The Mapping Cylinder ...... 81 4.2 The Double Mapping Cylinder ...... 84 4.3 Suspension. Homotopy Groups ...... 86 4.4 Loop Space ...... 89 viii Contents
4.5 Groups and Cogroups ...... 90 4.6 The Cofibre Sequence ...... 92 4.7 The Fibre Sequence ...... 97
5Cofibrations and Fibrations 101 5.1 The Homotopy Extension Property ...... 101 5.2 Transport ...... 107 5.3 Replacing a Map by a Cofibration ...... 110 5.4 Characterization of Cofibrations ...... 113 5.5 The Homotopy Lifting Property ...... 115 5.6 Transport ...... 119 5.7 Replacing a Map by a Fibration ...... 120
6 Homotopy Groups 121 6.1 The Exact Sequence of Homotopy Groups ...... 122 6.2 The Role of the Base Point ...... 126 6.3 Serre Fibrations ...... 129 6.4 The Excision Theorem ...... 133 6.5 The Degree ...... 135 6.6 The Brouwer Fixed Point Theorem ...... 137 6.7 Higher Connectivity ...... 141 6.8 Classical Groups ...... 146 6.9 Proof of the Excision Theorem ...... 148 6.10 Further Applications of Excision ...... 152
7 Stable Homotopy. Duality 159 7.1 A Stable Category ...... 159 7.2 Mapping Cones ...... 164 7.3 Euclidean Complements ...... 168 7.4 The Complement Duality Functor ...... 169 7.5 Duality ...... 175 7.6 Homology and Cohomology for Pointed Spaces ...... 179 7.7 Spectral Homology and Cohomology ...... 181 7.8 Alexander Duality ...... 185 7.9 Compactly Generated Spaces ...... 186
8 Cell Complexes 196 8.1 Simplicial Complexes ...... 197 8.2 Whitehead Complexes ...... 199 8.3 CW-Complexes ...... 203 8.4 Weak Homotopy Equivalences ...... 207 8.5 Cellular Approximation ...... 210 8.6 CW-Approximation ...... 211 Contents ix
8.7 Homotopy Classification ...... 216 8.8 Eilenberg–Mac Lane Spaces ...... 217
9 Singular Homology 223 9.1 Singular Homology Groups ...... 224 9.2 The Fundamental Group ...... 227 9.3 Homotopy ...... 228 9.4 Barycentric Subdivision. Excision ...... 231 9.5 Weak Equivalences and Homology ...... 235 9.6 Homology with Coefficients ...... 237 9.7 The Theorem of Eilenberg and Zilber ...... 238 9.8 The Homology Product ...... 241
10 Homology 244 10.1 The Axioms of Eilenberg and Steenrod ...... 244 10.2 Elementary Consequences of the Axioms ...... 246 10.3 Jordan Curves. Invariance of Domain ...... 249 10.4 Reduced Homology Groups ...... 252 10.5 The Degree ...... 256 10.6 The Theorem of Borsuk and Ulam ...... 261 10.7 Mayer–Vietoris Sequences ...... 265 10.8 Colimits ...... 270 10.9 Suspension ...... 273
11 Homological Algebra 275 11.1 Diagrams ...... 275 11.2 Exact Sequences ...... 279 11.3 Chain Complexes ...... 283 11.4 Cochain complexes ...... 285 11.5 Natural Chain Maps and Homotopies ...... 286 11.6 Chain Equivalences ...... 287 11.7 Linear Algebra of Chain Complexes ...... 289 11.8 The Functors Tor and Ext ...... 292 11.9 Universal Coefficients ...... 295 11.10 The Künneth Formula ...... 298
12 Cellular Homology 300 12.1 Cellular Chain Complexes ...... 300 12.2 Cellular Homology equals Homology ...... 304 12.3 Simplicial Complexes ...... 306 12.4 The Euler Characteristic ...... 308 12.5 Euler Characteristic of Surfaces ...... 311 x Contents
13 Partitions of Unity in Homotopy Theory 318 13.1 Partitions of Unity ...... 318 13.2 The Homotopy Colimit of a Covering ...... 321 13.3 Homotopy Equivalences ...... 324 13.4 Fibrations ...... 325
14 Bundles 328 14.1 Principal Bundles ...... 328 14.2 Vector Bundles ...... 335 14.3 The Homotopy Theorem ...... 342 14.4 Universal Bundles. Classifying Spaces ...... 344 14.5 Algebra of Vector Bundles ...... 351 14.6 Grothendieck Rings of Vector Bundles ...... 355
15 Manifolds 358 15.1 Differentiable Manifolds ...... 358 15.2 Tangent Spaces and Differentials ...... 362 15.3 Smooth Transformation Groups ...... 366 15.4 Manifolds with Boundary ...... 369 15.5 Orientation ...... 372 15.6 Tangent Bundle. Normal Bundle ...... 374 15.7 Embeddings ...... 379 15.8 Approximation ...... 383 15.9 Transversality ...... 384 15.10 Gluing along Boundaries ...... 388
16 Homology of Manifolds 392 16.1 Local Homology Groups ...... 392 16.2 Homological Orientations ...... 394 16.3 Homology in the Dimension of the Manifold ...... 396 16.4 Fundamental Class and Degree ...... 399 16.5 Manifolds with Boundary ...... 402 16.6 Winding and Linking Numbers ...... 403
17 Cohomology 405 17.1 Axiomatic Cohomology ...... 405 17.2 Multiplicative Cohomology Theories ...... 409 17.3 External Products ...... 413 17.4 Singular Cohomology ...... 416 17.5 Eilenberg–Mac Lane Spaces and Cohomology ...... 419 17.6 The Cup Product in Singular Cohomology ...... 422 17.7 Fibration over Spheres ...... 425 17.8 The Theorem of Leray and Hirsch ...... 427 Contents xi
17.9 The Thom Isomorphism ...... 431
18 Duality 438 18.1 The Cap Product ...... 438 18.2 Duality Pairings ...... 441 18.3 The Duality Theorem ...... 444 18.4 Euclidean Neighbourhood Retracts ...... 447 18.5 Proof of the Duality Theorem ...... 451 18.6 Manifolds with Boundary ...... 455 18.7 The Intersection Form. Signature ...... 457 18.8 The Euler Number ...... 461 18.9 Euler Class and Euler Characteristic ...... 464
19 Characteristic Classes 467 19.1 Projective Spaces ...... 468 19.2 Projective Bundles ...... 471 19.3 Chern Classes ...... 472 19.4 Stiefel–Whitney Classes ...... 478 19.5 Pontrjagin Classes ...... 479 19.6 Hopf Algebras ...... 482 19.7 Hopf Algebras and Classifying Spaces ...... 486 19.8 Characteristic Numbers ...... 491
20 Homology and Homotopy 495 20.1 The Theorem of Hurewicz ...... 495 20.2 Realization of Chain Complexes ...... 501 20.3 Serre Classes ...... 504 20.4 Qualitative Homology of Fibrations ...... 505 20.5 Consequences of the Fibration Theorem ...... 508 20.6 Hurewicz and Whitehead Theorems modulo Serre classes ..... 510 20.7 Cohomology of Eilenberg–Mac Lane Spaces ...... 513 20.8 Homotopy Groups of Spheres ...... 514 20.9 Rational Homology Theories ...... 518
21 Bordism 521 21.1 Bordism Homology ...... 521 21.2 The Theorem of Pontrjagin and Thom ...... 529 21.3 Bordism and Thom Spectra ...... 535 21.4 Oriented Bordism ...... 537
Bibliography 541 Symbols 551 Index 557
Chapter 1 Topological Spaces
In this chapter we collect the basic terminology about topological spaces and some elementary results (without proofs). I assume that the reader has some experience with point-set topology including the notion of compactness. We introduce a num- ber of examples and standard spaces that will be used throughout the book. Perhaps the reader has not met quotient spaces. Quotient spaces give precision to the in- tuitive concept of gluing and pasting. They comprise adjunction spaces, pushouts, attaching of spaces (in particular cells), orbit spaces of group actions. In the main text we deal with other topics: Mapping spaces and compact open topology, bun- dles, cell complexes, manifolds, partitions of unity, compactly generated spaces. Transformation groups are another topic of this chapter. Whenever you study a mathematical object you should consider its symmetries. In topology one uses, of course, continuous symmetries. They are called actions of a topological group on a space or transformation groups. In this chapter we assemble notions and results about the general topology of transformation groups. We use the material later for several purposes: • Important spaces like spheres, projective spaces and Grassmann manifolds have a high degree of symmetry which comes from linear algebra (matrix multiplication). • The fundamental group of a space has a somewhat formal definition. In the theory of covering spaces the fundamental group is exhibited as a symmetry group. This “hidden” symmetry, which is associated to a space, will influence several other of its geometric investigations. • The theory of fibre bundles and vector bundles makes essential use of the concept of a transformation group. Important information about a manifold is codified in its tangent bundle. We will apply the tools of algebraic topology to bundles (characteristic classes; classifying spaces). We should point out that large parts of algebraic topology can be generalized to the setting of transformation groups (equivariant topology). At a few occasions later we point out such generalizations.
1.1 Basic Notions
A topology on a set X is a set O of subsets of X, called open sets, with the properties: (1) The union of an arbitrary family of open sets is open. (2) The intersection of a finite family of open sets is open. (3) The empty set ; and X are open. A topological 2 Chapter 1. Topological Spaces space .X; O/ consists of a set X and a topology O on X. The sets in O are the open sets of the topological space .X; O/. We usually denote a topological space just by the underlying set X. A set A X is closed in .X; O/ if the complement X X A is open in .X; O/. Closed sets have properties dual to (1)–(3): The intersection of an arbitrary family of closed sets is closed; the union of a finite family of closed sets is closed; the empty set ; and X are closed. A subset B of a topology O is a basis of O if each U 2 O is a union of elements of B. (The empty set is the union of the empty family.) A subset S of O is a subbasis of O if the set of finite intersections of elements in S is a basis of O. (The space X is the intersection of the empty family.) A map f W X ! Y between topological spaces is continuous if the pre-image f 1.V / of each open set V of Y is open in X. Dually: A map is continuous if the pre-image of each closed set is closed. The identity id.X/W X ! X is always continuous, and the composition of continuous maps is continuous. Hence topological spaces and continuous maps form a category. We denote it by TOP. A homeomorphism f W X ! Y is a continuous map with a continuous inverse g W Y ! X. Spaces X and Y are homeomorphic if there exists a homeomorphism between them. A map f W X ! Y between topological spaces is open (closed)if the image of each open (closed) set is again open (closed). In the sequel we assume that a map between topological spaces is continuous if nothing else is specified or obvious. A set map is a map which is not assumed to be continuous at the outset. We fix a topological space X and a subset A. The intersection of the closed sets which contain A is denoted Ax and called closure of A in X. A set A is dense in X if Ax D X. The interior of A is the union of the open sets contained in A. We denote the interior by Aı. A point in Aı is an interior point of A. A subset is nowhere dense if the interior of its closure is empty. The boundary of A in X is Bd.A/ D Ax\ .X X A/. An open subset U of X which contains A is an open neighbourhood of A in X. A set B is a neighbourhood of A if it contains an open neighbourhood. A system of neighbourhoods of the point x is a neighbourhood basis of x if each neighbourhood of x contains one of the system. Given two topological spaces X and Y , a map f W X ! Y is said to be continu- ous at x 2 X if for each neighbourhood V of f.x/there exists a neighbourhood U of x such that f.U/ V ; it suffices to consider a neighbourhood basis of x and f.x/. Suppose O1 and O2 are topologies on X.IfO1 O2, then O2 is finer than O1 and O1 coarser than O2. The topology O2 is finer than O1 if and only if the identity .X; O2/ ! .X; O1/ is continuous. The set of all subsets of X is the finest topology; it is the discrete topology and the resulting space a discrete space. All maps f W X ! Y from a discrete space X are continuous. The coarsest topology on X consists of ; and X alone. If .Oj j j 2 J/is a family of topologies on X, then their intersection is a topology. 1.1. Basic Notions 3
We list some properties which a space X may have.
(T1) One-point subspaces are closed. (T2) Any two points have disjoint neighbourhoods. (T3) Given a point x 2 X and a closed subset A X not containing x, there exist disjoint neighbourhoods U of x and V of A.
(T4) Any two disjoint closed subsets have disjoint neighbourhoods. We say X satisfies the separation axiom Tj (or X is a Tj -space), if X has property Tj . The separation axioms are of a technical nature, but they serve the purpose of clarifying the concepts. A T2-space is called a Hausdorff space or separated. A space satisfying T1 and T3 is said to be regular. A space satisfying T1 and T4 is called normal.In a regular space, each neighbourhood of a point contains a closed neighbourhood. A space X is called completely regular if it is separated and for each x 2 X and ; 6D A X closed, x … A, there exists a continuous function f W X ! Œ0; 1 such that f.x/D 1 and f .A/ Df0g. A remarkable consequence of the separation property T4 is the existence of many real-valued continuous functions. The Urysohn existence theorem (1.1.1) shows that normal spaces are completely regular.
(1.1.1) Theorem (Urysohn). Let X be a T4-space and suppose that A and B are disjoint closed subsets of X. Then there exists a continuous function f W X ! Œ0; 1 with f .A/ f0g and f.B/ f1g.
(1.1.2) Theorem (Tietze). Let X be a T4-space and A X closed. Then each continuous map f W A ! Œ0; 1 has a continuous extension f W X ! Œ0; 1 . n A continuous map f W A ! R from a closed subset A of a T4-space X has a continuous extension to X. Many examples of topological spaces arise from metric spaces. Metric spaces are important in their own right. A metric d on a set X is a map d W X X ! Œ0; 1Œ with the properties: (1) d.x;y/ D 0 if and only if x D y. (2) d.x;y/ D d.y;x/ for all x;y 2 X. (3) d.x;z/ d.x;y/ C d.y;z/ for all x;y;z 2 X (triangle inequality). We call d.x;y/the distance between x and y with respect to the metric d.Ametric space .X; d/ consists of a set X and a metric d on X. Let .X; d/ be a metric space. The set U".x/ Dfy 2 X j d.x;y/ < "g is the "-neighbourhood of x. We call U X open with respect to d if for each x 2 X there exists ">0such that U".x/ U . The system Od of subsets U which are open with respect to d is a topology on X, the underlying topology of the metric space, and the "-neighbourhoods of all points are a basis for this topology. Subsets of the form U".x/ are open with respect to d. For the proof, let y 2 U".x/ 4 Chapter 1. Topological Spaces and 0<