## Topology and GeometryThe golden age of mathematics-that was not the age of Euclid, it is ours. C. J. KEYSER This time of writing is the hundredth anniversary of the publication (1892) of Poincare's first note on topology, which arguably marks the beginning of the subject of algebraic, or "combinatorial," topology. There was earlier scattered work by Euler, Listing (who coined the word "topology"), Mobius and his band, Riemann, Klein, and Betti. Indeed, even as early as 1679, Leibniz indicated the desirability of creating a geometry of the topological type. The establishment of topology (or "analysis situs" as it was often called at the time) as a coherent theory, however, belongs to Poincare. Curiously, the beginning of general topology, also called "point set topology," dates fourteen years later when Frechet published the first abstract treatment of the subject in 1906. Since the beginning of time, or at least the era of Archimedes, smooth manifolds (curves, surfaces, mechanical configurations, the universe) have been a central focus in mathematics. They have always been at the core of interest in topology. After the seminal work of Milnor, Smale, and many others, in the last half of this century, the topological aspects of smooth manifolds, as distinct from the differential geometric aspects, became a subject in its own right. |

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### Contents

Acknowledgments 1X Chapter | 1 |

Topological Spaces | 3 |

Subspaces | 8 |

Connectivity and Components | 10 |

Separation Axioms | 12 |

Nets MooreSmith Convergence | 14 |

Compactness | 18 |

Products | 22 |

Singular Homology | 219 |

The Cross Product | 220 |

Subdivision | 223 |

The MayerVietoris Sequence | 228 |

The Generalized Jordan Curve Theorem | 230 |

The BorsukUlam Theorem | 240 |

Simplicial Complexes | 245 |

Simplicial Maps | 250 |

Metric Spaces Again | 25 |

Existence of Real Valued Functions | 29 |

Locally Compact Spaces | 31 |

Paracompact Spaces | 35 |

Quotient Spaces | 39 |

Homotopy | 44 |

Topological Groups | 51 |

Convex Bodies | 56 |

The Baire Category Theorem | 57 |

Chapter II | 60 |

Differentiable Manifolds | 63 |

Differentiable Manifolds | 68 |

Local Coordinates | 71 |

Induced Structures and Examples | 72 |

Tangent Vectors and Differentials | 76 |

Sards Theorem and Regular Values | 80 |

Local Properties of Immersions and Submersions | 82 |

Vector Fields and Flows | 86 |

Tangent Bundles | 88 |

Embedding in Euclidean Space | 89 |

Tubular Neighborhoods and Approximations | 92 |

Classical Lie Groups ft | 101 |

Fiber Bundles ft | 106 |

Induced Bundles and Whitney Sums ft Ill 15 Transversality ft | 114 |

ThomPon try agin Theory ft | 118 |

Chapter III | 126 |

Fundamental Group | 127 |

The Fundamental Group | 132 |

Covering Spaces | 138 |

The Lifting Theorem | 143 |

The Action of 7tj on the Fiber | 146 |

Deck Transformations | 147 |

Properly Discontinuous Actions | 150 |

Classification of Covering Spaces | 154 |

The SeifertVan Kampen Theorem ft | 158 |

Remarks on SO3 ft | 164 |

Homology Theory 18 | 168 |

The Zeroth Homology Group | 172 |

Functorial Properties | 175 |

Homological Algebra | 177 |

Axioms for Homology | 182 |

Computation of Degrees | 190 |

CWComplexes | 197 |

Conventions for CWComplexes | 198 |

Cellular Homology | 200 |

Cellular Maps | 207 |

Products of CWComplexes ft | 211 |

Eulers Formula | 215 |

Homology of Real Projective Space | 217 |

The LefschetzHopf Fixed Point Theorem | 253 |

Chapter V | 255 |

Cohomology | 260 |

Differential Forms | 261 |

Integration of Forms | 265 |

Stokes Theorem | 267 |

Relationship to Singular Homology | 269 |

More Homological Algebra | 271 |

Universal Coefficient Theorems | 281 |

Excision and Homotopy | 285 |

de Rhams Theorem | 286 |

The de Rham Theory of CP | 292 |

Hopfs Theorem on Maps to Spheres | 297 |

Differential Forms on Compact Lie Groups Q | 304 |

Chapter VI | 315 |

A Sign Convention | 321 |

The Cup Product | 326 |

The Cap Product | 334 |

Classical Outlook on Duality | 338 |

The Orientation Bundle | 340 |

Duality Theorems | 348 |

Duality on Compact Manifolds with Boundary | 355 |

Applications of Duality | 359 |

Intersection Theory Q | 366 |

The Euler Class Lefschetz Numbers and Vector Fields | 378 |

The Gysin Sequence | 390 |

Lefschetz Coincidence Theory | 393 |

Steenrod Operations | 404 |

Construction of the Steenrod Squares | 412 |

StiefelWhitney Classes | 420 |

Plumbing | 426 |

Chapter VII | 430 |

The CompactOpen Topology | 437 |

HSpaces HGroups and HCogroups | 441 |

Homotopy Groups | 443 |

The Homotopy Sequence of a Pair | 445 |

Fiber Spaces | 450 |

Free Homotopy | 457 |

Classical Groups and Associated Manifolds | 463 |

The Hurewicz Theorem | 475 |

EilenbergMac Lane Spaces | 488 |

Obstruction Theory | 497 |

Obstruction Cochains and Vector Bundles | 511 |

Appendices | 519 |

Critical Values | 531 |

Bibliography | 541 |

549 | |

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### Common terms and phrases

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