## 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 |

LXV | 217 |

LXVI | 219 |

LXVII | 220 |

LXVIII | 222 |

LXIX | 225 |

LXX | 227 |

LXXI | 228 |

LXXII | 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 |

XLVI | 156 |

XLVII | 158 |

XLVIII | 163 |

XLIX | 165 |

L | 170 |

LI | 173 |

LII | 177 |

LIII | 179 |

LIV | 182 |

LV | 186 |

LVI | 189 |

LVII | 193 |

LVIII | 196 |

LIX | 197 |

LX | 201 |

LXI | 205 |

LXII | 207 |

LXIII | 210 |

LXIV | 213 |

LXXIII | 233 |

LXXIV | 236 |

LXXV | 242 |

LXXVI | 247 |

LXXVII | 251 |

LXXVIII | 252 |

LXXIX | 256 |

LXXX | 261 |

LXXXI | 263 |

LXXXII | 268 |

LXXXIII | 272 |

LXXXIV | 277 |

LXXXV | 281 |

LXXXVI | 284 |

LXXXVII | 289 |

LXXXVIII | 291 |

LXXXIX | 295 |

XC | 299 |

XCI | 303 |

XCII | 306 |

XCIII | 313 |

XCIV | 317 |

XCV | 320 |

XCVI | 324 |

XCVII | 325 |

XCVIII | 328 |

XCIX | 332 |

C | 334 |

CI | 339 |

CII | 343 |

CIII | 346 |

CIV | 350 |

CV | 355 |

CVI | 359 |

CVII | 365 |

CVIII | 367 |

CIX | 369 |

CX | 370 |

CXI | 371 |

CXII | 373 |

CXIII | 375 |

CXIV | 378 |

CXV | 380 |

CXVI | 385 |

CXVII | 387 |

CXVIII | 389 |

CXIX | 391 |

CXX | 393 |

CXXI | 397 |

419 | |

421 | |

425 | |

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### Common terms and phrases

1-chain abelian group Algebraic analytic apply assume basis boundary called Chapter choice circle cohomology commutes compact complement complex connected connected components constant construct contained continuous mapping coordinate Corollary corresponding covering curve define definition denoted determines differential disjoint disk edges element equation equivalent exact example Exercise fact finite fixed follows formula function fundamental group G-covering given gives homology group homomorphism homotopy identified identity independent integral interval isomorphism Lemma lifting linear locally manifold Mayer-Vietoris means meromorphic neighborhood Note open set orientation particular path plane Problem projection proof Proposition prove rectangle relation restriction Riemann surface sequence Show sides simply singularities space sphere subgroup subset Suppose takes theorem Theory topology trivial union unique vanishes vector field Verify vertices winding number write zero

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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.