## Algebraic TopologyIn most major universities one of the three or four basic first-year graduate mathematics courses is algebraic topology. This introductory text is suitable for use in a course on the subject or for self-study, featuring broad coverage and a readable exposition, with many examples and exercises. The four main chapters present the basics: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally. The author emphasizes the geometric aspects of the subject, which helps students gain intuition. A unique feature is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers. |

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### Contents

II | ix |

III | 1 |

IV | 4 |

V | 6 |

VI | 10 |

VII | 17 |

IX | 21 |

X | 25 |

XLVIII | 229 |

XLIX | 235 |

L | 245 |

LI | 248 |

LII | 257 |

LIII | 264 |

LIV | 277 |

LV | 288 |

XI | 30 |

XII | 36 |

XIII | 37 |

XIV | 39 |

XV | 46 |

XVI | 52 |

XVII | 56 |

XVIII | 59 |

XIX | 66 |

XX | 79 |

XXI | 83 |

XXII | 93 |

XXIII | 98 |

XXIV | 100 |

XXV | 104 |

XXVI | 106 |

XXVII | 109 |

XXVIII | 124 |

XXIX | 130 |

XXX | 133 |

XXXI | 145 |

XXXII | 149 |

XXXIII | 156 |

XXXIV | 158 |

XXXV | 162 |

XXXVI | 165 |

XXXVII | 173 |

XXXVIII | 181 |

XL | 186 |

XLII | 193 |

XLIII | 202 |

XLIV | 207 |

XLV | 214 |

XLVI | 220 |

XLVII | 226 |

LVI | 299 |

LVII | 307 |

LVIII | 317 |

LIX | 323 |

LX | 333 |

LXI | 335 |

LXII | 336 |

LXIII | 342 |

LXIV | 344 |

LXV | 348 |

LXVI | 356 |

LXVIII | 362 |

LXIX | 371 |

LXX | 380 |

LXXI | 389 |

LXXII | 401 |

LXXIII | 406 |

LXXIV | 411 |

LXXV | 417 |

LXXVI | 423 |

LXXVII | 425 |

LXXVIII | 427 |

LXXIX | 444 |

LXXX | 448 |

LXXXI | 452 |

LXXXII | 456 |

LXXXIII | 462 |

LXXXIV | 466 |

LXXXV | 471 |

LXXXVI | 483 |

LXXXVII | 515 |

LXXXVIII | 529 |

535 | |

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### Common terms and phrases

abelian action algebra algebraic topology apply associated attaching basepoint basis boundary bundle called cell cellular chain complex circle closed coefficients cohomology commutative compact composition connected consider consisting construction contained continuous copies corresponding covering space cup product CW complex defined definition deformation retracts diagram dimension direct edges element example exercise extend fact factors fiber finite fixed formula fundamental given gives graph hence homology groups homomorphism homotopy equivalence identified identity implies inclusion induces injective isomorphism lift long exact sequence loop multiplication natural neighborhood obtained orientable pair path path-connected polynomial preceding projection Proof Proposition prove quotient reduced relation relative represented restriction result ring says simplicial singular structure subcomplex subgroup subspace surjective suspension theorem theory topology trivial union unique universal vertices zero