## Algebraic Topology: A First CourseTo the Teacher. This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the re lations of these ideas with other areas of mathematics. Rather than choosing one point of view of modem topology (homotopy theory, simplicial complexes, singular theory, axiomatic homology, differ ential topology, etc.), we concentrate our attention on concrete prob lems in low dimensions, introducing only as much algebraic machin ery as necessary for the problems we meet. This makes it possible to see a wider variety of important features of the subject than is usual in a beginning text. The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topol ogists-without, we hope, discouraging budding topologists. We also feel that this approach is in better harmony with the historical devel opment of the subject. What would we like a student to know after a first course in to pology (assuming we reject the answer: half of what one would like the student to know after a second course in topology)? Our answers to this have guided the choice of material, which includes: under standing the relation between homology and integration, first on plane domains, later on Riemann surfaces and in higher dimensions; wind ing numbers and degrees of mappings, fixed-point theorems; appli cations such as the Jordan curve theorem, invariance of domain; in dices of vector fields and Euler characteristics; fundamental groups |

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### Contents

III | 3 |

IV | 7 |

V | 10 |

VI | 17 |

VII | 23 |

VIII | 27 |

IX | 33 |

X | 35 |

LXIV | 217 |

LXV | 219 |

LXVI | 220 |

LXVII | 222 |

LXVIII | 225 |

LXIX | 227 |

LXX | 228 |

LXXI | 231 |

XI | 38 |

XII | 42 |

XIII | 43 |

XIV | 48 |

XV | 49 |

XVI | 53 |

XVII | 56 |

XVIII | 59 |

XIX | 63 |

XX | 65 |

XXI | 68 |

XXII | 72 |

XXIII | 78 |

XXIV | 82 |

XXV | 85 |

XXVI | 89 |

XXVII | 91 |

XXVIII | 95 |

XXIX | 97 |

XXX | 101 |

XXXI | 102 |

XXXII | 106 |

XXXIII | 113 |

XXXIV | 121 |

XXXV | 123 |

XXXVI | 127 |

XXXVII | 130 |

XXXVIII | 131 |

XXXIX | 137 |

XL | 140 |

XLI | 144 |

XLII | 147 |

XLIII | 151 |

XLIV | 153 |

XLV | 156 |

XLVI | 158 |

XLVII | 163 |

XLVIII | 165 |

XLIX | 170 |

L | 173 |

LI | 177 |

LII | 179 |

LIII | 182 |

LIV | 186 |

LV | 189 |

LVI | 193 |

LVII | 196 |

LVIII | 197 |

LIX | 201 |

LX | 205 |

LXI | 207 |

LXII | 210 |

LXIII | 213 |

LXXII | 233 |

LXXIII | 236 |

LXXIV | 242 |

LXXV | 247 |

LXXVI | 251 |

LXXVII | 252 |

LXXVIII | 256 |

LXXIX | 261 |

LXXX | 263 |

LXXXI | 268 |

LXXXII | 272 |

LXXXIII | 277 |

LXXXIV | 281 |

LXXXV | 284 |

LXXXVI | 289 |

LXXXVII | 291 |

LXXXVIII | 295 |

LXXXIX | 299 |

XC | 303 |

XCI | 306 |

XCII | 313 |

XCIII | 317 |

XCIV | 320 |

XCV | 324 |

XCVI | 325 |

XCVII | 328 |

XCVIII | 332 |

XCIX | 334 |

C | 339 |

CI | 343 |

CII | 346 |

CIII | 350 |

CIV | 355 |

CV | 359 |

CVI | 365 |

CVII | 367 |

CVIII | 369 |

CIX | 370 |

CX | 371 |

CXI | 373 |

CXII | 375 |

CXIII | 378 |

CXIV | 380 |

CXV | 385 |

CXVI | 387 |

CXVII | 389 |

CXVIII | 391 |

CXIX | 393 |

CXX | 397 |

419 | |

421 | |

425 | |

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### Common terms and phrases

1-cycle Algebraic base point boundary calculation chain Chapter circle closed 1-form closed paths cocycle cohomology commutes compact support complement connected components construct contained continuous mapping coordinate Corollary corresponding covering map curve def,ne define deg(D degree denoted determines diffeomorphic differential dim(L(D disjoint union disk divisor edges element endpoints equation equivalent exact sequence Exercise fact finite number follows formula free abelian group fundamental group G-covering given group G homology group homomorphism homotopy integral intersection interval isomorphism Jordan curve theorem Lemma linear mapping locally constant function loop manifold Mayer-Vietoris meromorphic function morphism neighborhood nonzero open sets open subset orientation path-connected polynomial Problem proof Proposition prove quotient restriction Show simply connected singularities sphere subgroup subspace Suppose surjective takes theorem topological space topology torus triangulation trivial unique universal covering vanishes vector field vector space Verify vertices winding number zero

### Popular passages

Page v - The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topologists — without, we hope, discouraging budding topologists.