Table Of Contents:
Preface v
Notations xv

Part 1. Simplicial Techniques and Homology

Introduction 3(12)

The Work of Poincare 15(21)

Introduction 15(1)

Poincare's First Paper: Analysis Situs 16(12)

Heegaard's Criticisms and the First Two Complements a l' Analysis Situs 28(8)

The Build-Up of ``Classical'' Homology 36(24)

The Successors of Poincare 36(1)

The Evolution of Basic Concepts and Problems 37(6)

The Invariance Problem 43(6)

Duality and Intersection Theory on Manifolds 49(6)

The Notion of ``Manifold'' 49(1)

Computation of Homology by Blocks 50(1)

Poincare Duality for Combinatorial Manifolds 50(1)

Intersection Theory for Combinatorial Manifolds 51(4)

Homology of Products of Cell Complexes 55(1)

Alexander Duality and Relative Homology 56(4)

The Beginnings of Differential Topology 60(7)

Global Properties of Differential Manifolds 60(2)

The Triangulation of C1 Manifolds 62(1)

The Theorems of de Rham 62(5)

The Various Homology and Cohomology Theories 67(100)

Introduction 67(1)

Singular Homology versus Cech Homology and the Concept of Duality 68(10)

Cohomology 78(3)

Products in Cohomology 81(4)

The Cup Product 81(3)

The Functional Cup Product 84(1)

The Cap Product 84(1)

The Growth of Algebraic Machinery and the Forerunners of Homological Algebra 85(20)

Exact Sequences 85(5)

The Functors ⊗ and Tor 90(2)

The Kunneth Formula for Chain Complexes 92(1)

The Functors Hom and Ext 93(3)

The Birth of Categories and Functors 96(2)

Chain Homotopies and Chain Equivalences 98(2)

Acyclic Models and the Eilenberg-Zilber Theorem 100(2)

Applications to Homology and Cohomology of Spaces: Cross Products and Slant Products 102(3)

Identifications and Axiomatizations 105(15)

Comparison of Vietoris, Cech, and Alexander-Spanier Theories 105(2)

The Axiomatic Theory of Homology and Cohomology 107(6)

Cohomology of Smooth Manifolds 113(2)

Cubical Singular Homology and Cohomology 115(1)

Leray's 1945 Paper 115(5)

Sheaf Cohomology 120(27)

Homology with Local Coefficients 120(3)

The Concept of Sheaf 123(5)

Sheaf Cohomology 128(4)

Spectral Sequences 132(6)

Applications of Spectral Sequences to Sheaf Cohomology 138(3)

Coverings and Sheaf Cohomology 141(2)

Borel-Moore Homology 143(4)

Homological Algebra and Category Theory 147(20)

Homological Algebra 147(2)

Dual Categories 149(2)

Representable Functors 151(4)

Abelian Categories 155(6)

Part 2. The First Applications of Simplicial Methods and of Homology

Introduction 161(6)

The Concept of Degree 167(15)

The Work of Brouwer 167(2)

The Brouwer Degree 169(4)

Later Improvements and Variations 173(7)

Homological Interpretation of the Degree 173(2)

Order of a Point with Respect to a Hypersurface; the Kronecker Integral and the Index of a Vector Field 175(1)

Linking Coefficients 176(2)

Localization of the Degree 178(2)

Applications of the Degree 180(2)

Dimension Theory and Separation Theorems 182(15)

The Invariance of Dimension 182(1)

The Invariance of Domain 183(2)

The Jordan-Brouwer Theorem 185(4)

Lebesgue's Note 185(1)

Brouwer's First Paper on the Jordan-Brouwer Theorem 185(2)

Brouwer's Second Paper on the Jordan-Brouwer Theorem 187(2)

The No Separation Theorem 189(2)

The Notion of Dimension for Separable Metric Spaces 191(3)

Later Developments 194(3)

Fixed Points 197(7)

The Theorems of Brouwer 197(1)

The Lefschetz Formula 198(3)

The Index Formula 201(3)

Local Homological Properties 204(10)

Local Invariants 204(4)

Local Homology Groups and Local Betti Numbers 204(2)

Application to the Local Degree 206(1)

Later Developments 206(1)

Phragmen-Brouwer Theorems and Unicoherence 207(1)

Homological and Cohomological Local Connectedness 208(2)

Duality in Manifolds and Generalized Manifolds 210(4)

Fundamental Classes and Duality 210(1)

Duality in Generalized Manifolds 211(3)

Quotient Spaces and Their Homology 214(18)

The Notion of Quotient Space 214(2)

Collapsing and Identifications 216(4)

Collapsing 216(1)

Cones 216(1)

Suspension 217(1)

Wedge and Smash Product 217(1)

Join of Two Spaces 218(1)

Doubling 218(1)

Connected Sums 219(1)

Attachments and CW-Complexes 220(3)

The Mapping Cylinder 220(1)

The Mapping Cone 220(1)

The CW-Complexes 221(2)

Applications: I. Homology of Grassmannians, Quadrics, and Stiefel Manifolds 223(4)

Homology of Projective Spaces 223(1)

Homology of Grassmannians 224(1)

Homology of Quadrics and Stiefel Manifolds 225(2)

Applications: II. The Morse Inequalities 227(5)

Homology of Groups and Homogeneous Spaces 232(17)

The Homology of Lie Groups 232(2)

H-Spaces and Hopf Algebras 234(8)

Hopf's Theorem 234(4)

Samelson's Theorem and Pontrjagm Product 238(3)

Interpretation of the Rank in Cohomology 241(1)

Remarks 242(1)

Action of Transformation Groups on Homology 242(7)

Complexes with Automorphisms 242(2)

The Franz-Reidemeister Torsion 244(2)

Fixed Points of Periodic Automorphisms 246(3)

Applications of Homology to Geometry and Analysis 249(44)

Applications to Algebraic Geometry 249(8)

Early Applications 249(2)

The Work of Lefschetz 251(2)

The Triangulation of Algebraic Varieties 253(1)

The Hodge Theory 254(3)

Applications to Analysis 257(5)

Fixed Point Theorems 257(3)

The Leray-Schauder Degree 260(2)

The Calculus of Variations in the Large (Morse Theory) 262(31)

Part 3. Homotopy and its Relation to Homology

Introduction 273(20)

Fundamental Group and Covering Spaces 293(18)

Covering Spaces 293(3)

The Theory of Covering Spaces 296(5)

Computation of Fundamental Groups 301(4)

Elementary Properties 301(1)

Fundamental Groups of Simplicial Complexes 301(1)

Covering Spaces of Complexes 302(1)

The Seifert-van Kampen Theorem 302(2)

Fundamental Group and One-Dimensional Homology Group 304(1)

Examples and Applications 305(6)

Fundamental Groups of Graphs 305(1)

The ``Gruppenbild'' 306(1)

Fundamental Group of a H-Space 307(1)

Poincare Manifolds 307(1)

Knots and Links 307(4)

Elementary Notions and Early Results in Homotopy Theory 311(74)

The Work of H. Hopf 311(9)

Brouwer's Conjecture 311(3)

The Hopf Invariant 314(3)

Generalizations to Maps from S2k-1 into Sk 317(3)

Basic Notions in Homotopy Theory 320(10)

Homotopy and Extensions 320(1)

Retracts and Extensions 321(2)

Homotopy Type 323(3)

Retracts and Homotopy 326(3)

Fixed Points and Retracts 329(1)

The Lusternik-Schnirelmann Category 329(1)

Homotopy Groups 330(9)

The Hurewicz Definition 330(3)

Elementary Properties of Homotopy Groups 333(1)

Suspensions and Loop Spaces 334(2)

The Homotopy Suspension 336(1)

Whitehead Products 337(1)

Change of Base Points 338(1)

First Relations Between Homotopy and Homology 339(8)

The Hurewicz Homomorphism 339(2)

Application to the Hopf Classification Problem 341(1)

Obstruction Theory 342(5)

Relative Homotopy and Exact Sequences 347(9)

Relative Homotopy Groups 347(3)

The Exact Homotopy Sequence 350(2)

Triples and Triads 352(1)

The Barratt-Puppe Sequence 353(2)

The Relative Hurewicz Homomorphism 355(1)

The First Whitehead Theorem 355(1)

Homotopy Properties of CW-Complexes 356(13)

Aspherical Spaces 356(2)

The Second Whitehead Theorem 358(2)

Lemmas on Homotopy in Relative CW-Complexes 360(2)

The Homotopy Excision Theorem 362(2)

The Freudenthal Suspension Theorems 364(2)

Realizability of Homotopy Groups 366(3)

Spaces Having the Homotopy Type of CW-Complexes 369(1)

Simple Homotopy Type 369(16)

Formal Deformations 369(3)

The Whitehead Torsion 372(13)

Fibrations 385(36)

Fibers and Fiber Spaces 385(13)

From Vector Fields to Fiber Spaces 385(2)

The Definition of (Locally Trivial) Fiber Spaces 387(2)

Basic Properties of Fiber Spaces 389(9)

Homotopy Properties of Fibrations 398(23)

Covering Homotopy and Fibrations 398(4)

Fiber Spaces and Fibrations 402(4)

The Homotopy Exact Sequence of a Fibration 406(3)

Applications to Computations of Homotopy Groups 409(2)

Classifying Spaces: I. The Whitney-Steenrod Theorems 411(3)

Classifying Spaces: II. Later Improvements 414(3)

Classifying Spaces: III. The Milnor Construction 417(2)

The Classification of Principal Fiber Spaces with Base Space Sn 419(2)

Homology of Fibrations 421(32)

Characteristic Classes 421(14)

The Stiefel Classes 421(1)

Whitney's Work 422(4)

Pontrjagin Classes 426(4)

Chern Classes 430(2)

Later Results 432(3)

The Gysin Exact Sequence 435(4)

The Spectral Sequences of a Fibration 439(8)

The Leray Cohomological Spectral Sequence of a Fiber Space 439(3)

The Transgression 442(1)

The Serre Spectral Sequences 443(4)

Applications to Principal Fiber Spaces 447(6)

Sophisticated Relations between Homotopy and Homology 453(57)

Homology and Cohomology of Discrete Groups 453(12)

The Second Homology Group of a Simplicial Complex 453(2)

The Homology of Aspherical Simplicial Complexes 455(2)

The Eilenberg Groups 457(1)

Homology and Cohomology of Groups 458(5)

Application to Covering Spaces 463(2)

Postnikov Towers and Eilenberg-Mac Lane Fibers 465(9)

The Eilenberg-Mac Lane Invariant 465(1)

The Postnikov Invariants 466(3)

Fibrations with Eilenberg-Mac Lane Fibers 469(3)

The Homology Suspension 472(2)

The Homology of Eilenberg-Mac Lane Spaces 474(4)

The Topological Approach 474(1)

The Bar Construction 475(2)

The Cartan Constructions 477(1)

Serre's g-Theory 478(6)

Definitions 478(3)

The Absolute g-Isomorphism Hurewicz Theorem 481(2)

The Relative g-Isomorphism Hurewicz Theorem 483(1)

The First Whitehead g-Theorem 483(1)

The Computation of Homotopy Groups of Spheres 484(12)

Serre's Finiteness Theorem for Odd-Dimensional Spheres 484(4)

Serre's Finiteness Theorem for Even-Dimensional Spheres 488(1)

Wedges of Spheres and Homotopy Operations 489(3)

Freudenthal Suspension, Hopf Invariant, and James Exact Sequence 492(3)

The Localization of Homotopy Groups 495(1)

The Explicit Computation of the πn+k(Sn) for k > 0 495(1)

The Computation of Homotopy Groups of Compact Lie Groups 496(14)

Serre's Method 496(2)

Bott's Periodicity Theorems 498(10)

Later Developments 508(2)

Cohomology Operations 510(45)

The Steenrod Squares 510(5)

Mappings of Spheres and Cup-Products 510(1)

The Construction of the Steenrod Squares 511(4)

The Steenrod Reduced Powers 515(8)

New Definition of the Steenrod Squares 515(2)

The Steenrod Reduced Powers: First Definition 517(2)

The Steenrod Reduced Powers: Second Definition 519(4)

Cohomology Operations 523(10)

Cohomology Operations and Eilenberg-Mac Lane Spaces 523(2)

The Cohomology Operations of Type (q, n, II, F2) 525(2)

The Relations between the Steenrod Squares 527(1)

The Relations between the Steenrod Reduced Powers, and the Steenrod Algebra 528(4)

The Pontrjagin p-th Powers 532(1)

Applications of Steenrod's Squares and Reduced Powers 533(12)

The Steenrod Extension Theorem 533(4)

Steenrod Squares and Stiefel-Whitney Classes 537(4)

Application to Homotopy Groups 541(3)

Nonexistence Theorems 544(1)

Secondary Cohomology Operations 545(6)

The Notion of Secondary Cohomology Operations 545(1)

General Constructions 546(2)

Special Secondary Cohomology Operations 548(1)

The Hopf Invariant Problem 549(2)

Consequences of Adams' Theorem 551(1)

Cohomotopy Groups 551(4)

Cohomotopy Sets 552(1)

Cohomotopy Groups 553(2)

Generalized Homology and Cohomology 555(57)

Cobordism 555(25)

The Work of Pontrjagin 555(1)

Transversality 556(2)

Thom's Basic Construction 558(1)

Homology and Homotopy of Thom Spaces 559(2)

The Realization Problem 561(2)

Smooth Classes in Simplicial Complexes 563(1)

Unoriented Cobordism 564(10)

Oriented Cobordism 574(1)

Later Developments 575(5)

First Applications of Cobordism 580(18)

The Riemann-Roch-Hirzebruch Theorem 580(1)

The Arithmetic Genus 580(1)

The Todd Genus 581(1)

Divisors and Line Bundles 582(2)

The Riemann-Roch Problem 584(1)

Virtual Genus and Arithmetic Genus 585(1)

The Introduction of Sheaves 586(1)

The Sprint 587(4)

The Grand Finale 591(4)

Exotic Spheres 595(3)

The Beginnings of K-Theory 598(5)

The Grothendieck Groups 598(3)

Riemann-Roch Theorems for Differentiable Manifolds 601(2)

S-Duality 603(3)

Spectra and Theories of Generalized Homology and Cohomology 606(6)

K-Theory and Generalized Cohomology 606(1)

Spectra 607(1)

Spectra and Generalized Cohomology 608(2)

Generalized Homology and Stable Homotopy 610(2)
Bibliography 612(21)
Index of Cited Names 633(6)
Subject Index 639