## Algebraic Topology: An Intuitive ApproachThe single most difficult thing one faces when one begins to learn a new branch of mathematics is to get a feel for the mathematical sense of the subject. The purpose of this book is to help the aspiring reader acquire this essential common sense about algebraic topology in a short period of time. To this end, Sato leads the reader through simple but meaningful examples in concrete terms. Moreover, results are not discussed in their greatest possible generality, but in terms of the simplest and most essential cases. In response to suggestions from readers of the original edition of this book, Sato has added an appendix of useful definitions and results on sets, general topology, groups and such. He has also provided references. Topics covered include fundamental notions such as homeomorphisms, homotopy equivalence, fundamental groups and higher homotopy groups, homology and cohomology, fiber bundles, spectral sequences and characteristic classes. Objects and examples considered in the text include the torus, the Mobius strip, the Klein bottle, closed surfaces, cell complexes and vector bundles. |

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### Contents

Homeomorphisms and Homotopy Equivalences | 1 |

Topological Spaces and Cell Complexes | 9 |

Homotopy | 17 |

Homology Groups of Cell Complexes | 39 |

Cohomology | 55 |

The Universal Coefficient Theorem | 61 |

Fiber Bundles and Vector Bundles | 69 |

Spectral Sequences | 81 |

### Common terms and phrases

abelian group algebraic topology attaching map attaching space base space BO(n boundary operators BU(n bundle map cell complex chain complex CHAPTER characteristic classes classifying space coefficient group cohomology spectral sequence compute the homology constant map continuous maps cup product cyclic group define DEFINITION denote direct sum element exact couple exact sequence EXAMPLE fiber bundle FIGURE fundamental group geometrical object GR(n Grassmann manifold group G groups hp(X Hence homeomorphic homology axioms homology theory homotopy class homotopy equivalence homotopy set homotopy type hp+l(X identity map In,dln incidence number inclusion map induced isomorphism map f mathematics Mobius strip n-dimensional vector bundle natural number p-th homology group product space projective plane P2(R PROOF PROPOSITION quotient space real projective plane satisfy the following Serre's simplex simplicial complex simply connected subset subspace topological pair topological space torus total space triangulation universal coefficient theorem vector bundle write