## Algebraic Topology from a Homotopical ViewpointThe authors present introductory material in algebraic topology from a novel point of view in using a homotopy-theoretic approach. This carefully written book can be read by any student who knows some topology, providing a useful method to quickly learn this novel homotopy-theoretic point of view of algebraic topology. |

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### Contents

FUNCTION SPACES | 2 |

12 CompactOpen Topology | 3 |

13 The Exponential Law | 4 |

CONNECTEDNESS AND ALGEBRAIC INVARIANTS | 9 |

22 Homotopy Classes | 10 |

23 Topological Groups | 13 |

24 Homotopy of Mappings of the Circle into Itself | 15 |

25 The Fundamental Group | 28 |

82 Projections and Vector Bundles | 268 |

83 Grassmann Manifolds and Universal Bundles | 271 |

84 Classification of Vector Bundles of Finite Type | 276 |

85 Classification of Vector Bundles over Paracompact Spaces | 279 |

𝐾THEORY | 289 |

92 Definition of 𝐾𝐵 | 292 |

93 𝐾𝐵 and Stable Equivalence of Vector Bundles | 295 |

94 Representations of 𝐾B and 𝐾𝐵 | 299 |

26 The fundamental Group of the Circle | 41 |

27 𝐻Spaces | 45 |

28 Loop Spaces | 48 |

29 𝐻COSPACES | 50 |

210 Suspensions | 53 |

HOMOTOPY GROUPS | 59 |

32 The Seifertvan Kampen Theorem | 63 |

33 Homotopy Sequences I | 72 |

34 Homotopy Groups | 80 |

35 Homotopy Sequences II | 84 |

HOMOTOPY EXTENSION AND LIFTING PROPERTIES | 89 |

42 Some Results on Cofibrations | 95 |

43 Fibrations | 101 |

44 Pointed and Unpointed Homotopy Classes | 119 |

45 Locally Trivial Bundles | 125 |

46 Classification of Covering Maps over Paracompact Spaces | 138 |

CWCOMPLEXES AND HOMOLOGY | 149 |

52 Infinite Symmetric Products | 167 |

53 Homology Groups | 176 |

HOMOTOPY PROPERTIES OF CWCOMPLEXES | 189 |

62 Homotopy Excision and Related Results | 193 |

63 Homotopy Properties of the Moore Spaces | 201 |

64 Homotopy Properties of the EilenbergMac Lane Spaces | 217 |

COHOMOLOGY GROUPS AND RELATED TOPICS | 227 |

72 Multiplication in Cohomology | 238 |

73 Cellular Homology and Cohomology | 243 |

74 Exact Sequences in Homology and Cohomology | 252 |

VECTOR BUNDLES | 259 |

95 Bott Periodicity and Applications | 302 |

ADAMS OPERATIONS AND APPLICATIONS | 309 |

102 The Splitting Principle | 313 |

103 Normed Algebras | 315 |

104 Division Algebras | 317 |

105 Multiplicative Structures on ℝⁿ and on 𝕊ⁿ¹ | 319 |

106 The Hopf Invariant | 321 |

RELATIONS BETWEEN COHOMOLOGY AND VECTOR BUNDLES | 331 |

111 Contractibility of 𝕊 | 332 |

112 Description of 𝐾ℤ21 | 334 |

113 Classification of Real Line Bundles | 337 |

114 Description of 𝐾ℤ2 | 340 |

115 Classification of Complex Line Bundles | 343 |

116 Characteristic Classes | 345 |

117 Thom Isomorphism and Gysin Sequence | 349 |

118 Construction of Characteristic Classes and Applications | 366 |

COHOMOLOGY THEORIES AND BROWN REPRESENTABILITY | 383 |

122 Brown Representability Theorem | 394 |

123 Spectra | 406 |

PROOF OF THE DOLDTHOM THEOREM | 421 |

A2 Symmetric Products | 431 |

A3 Proof of the DoldThom Theorem | 434 |

PROOF OF THE BOTT PERIODICITY THEOREM | 437 |

B2 Proof of the Bott Periodicity Theorem | 440 |

457 | |

SYMBOLS | 463 |

467 | |

### Other editions - View all

Algebraic Topology from a Homotopical Viewpoint Marcelo Aguilar,Samuel Gitler,Carlos Prieto Limited preview - 2008 |

Algebraic Topology from a Homotopical Viewpoint Marcelo Aguilar,Samuel Gitler,Carlos Prieto No preview available - 2011 |

Algebraic Topology from a Homotopical Viewpoint Marcelo Aguilar,Samuel Gitler,Carlos Prieto No preview available - 2013 |

### Common terms and phrases

abelian group algebra axiom base point bijection bundle of dimension canonical inclusion cells chapter coeﬃcients cofibration cohomology groups cohomology theory colimit commutative diagram compact composite consider construction continuous map Corollary countable covering map CW-complex deﬁne Definition deformation denote element epimorphism Exercise exists a homotopy fiber ﬁnite ﬁrst function functor fundamental group Gauss map given Hausdorff homology homomorphism homotopy classes homotopy groups homotopy H homotopy sequence homotopy type Hq(X Hurewicz inﬁnite inverse K-theory Lane spaces Lemma Let f line bundle locally trivial bundle loop map f map g monomorphism Moreover multiplication nullhomotopic obtain pair paracompact space path connected pointed CW-complex pointed map pointed space Proof Proposition Prove quasifibration quotient map real vector bundle reduced cohomology respectively retraction satisfy Sn_1 structure subset subspace Suppose symmetric product Theorem Thom topological space unique vector bundle vector space weak homotopy equivalence

### Popular passages

Page xxv - This slogan should be taken literally, but not naively: it means that we can apply intuitions from the category of topological spaces and continuous maps to the category Top, and hence to logic. It does not mean that any space is a topos, in the naive sense of "is".

Page 459 - JL Kelley, General topology (D. Van Nostrand Co., Inc., New York, 1955).