## Algebraic topology: a first course |

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This is an expanded and much improved revision of Greenberg's Lectures on Algebraic Topology (Benjamin 1967), Harper adding 76 pages to the original, most of which remains intact in this version.

Greenberg's book was most notable for its emphasis on the Eilenberg-Steenrod axioms for any homology theory and for the verification of those axioms for the obviously invariant singular homology theory. Using those those results that were now theorems, Greenberg showed how to calculate the homology groups of finite cell complexes (and more generally of a space obtained by adjunction from a known space). He thus obtained all the classical results for spheres, compact surfaces, real, complex and quaternionic projective spaces, lens spaces etc. without going through the tedious method of simplicial complexes. He was able similarly to prove the well-known duality theorems for manifolds and the Lefschetz Fixed Point Theorem, following ideas of Dold. His book began with the basic theory of the fundamental group and covering spaces; he defined the higher homotopy groups and proved they are abelian, but did not go further into that theory.

Greenberg's book heavily emphasized the algebraic aspect of algebraic topology. Harper's additions contributed a more geometric flavor to the development, adding many examples, figures and exercises to balance the algebra. Harper also provided slicker proofs of a few of the theorems in the original, and added lots of new material not previously discussed (such as about knots).

The result is a nicely balanced presentation of a branch of mathematics that began toward the end of the 19th century and has had a spectacular development ever since.

### Contents

Loop Spaces and Higher Homotopy Groups | 32 |

Singular Homology Theory | 37 |

Introduction to Part II | 39 |

Copyright | |

24 other sections not shown