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1. Algebraic Topology (Volume 0)by Edwin H. Spanier | |

Paperback: 548
Pages
(1994-12-06)
list price: US$79.95 -- used & new: US$40.00 (price subject to change: see help) Asin: 0387944265Average Customer Review:
Canada | United Kingdom | Germany | France | Japan | |

Editorial Review
After a brief introduction to set theory, general topology, and algebra, homotopy and the fundamental group are covered in Chapter 1. Categories and functors are defined, and some examples are given, but the reader will have to consult the literature for an in-depth discussion. Homotopy is introduced as an equivalence class of maps between topological pairs. Fixing a base point allows the author to define H-spaces, but he does not motivate the real need for using pointed spaces, namely as a way of obtaining the composition law for the loops in the fundamental group. By suitable use of the reduced join, reduced product, and reduced suspension, the author shows how to obtain H-groups and H co-groups. The fundamental group is defined in the last section of the chapter, and the author does clarify the non-uniqueness of the fundamental group based at different points of a path-connected space. Covering spaces and fibrations are discussed in the next chapter. The author does a fairly good job of discussing these, and does a very good job of motivating the definition of a fiber bundle as a generalized covering space where the "fiber" is not discrete. The fundamental group is used to classify covering spaces. In chapter 3 the author gets down to the task of computing the fundamental group of a space using polyhedra. Although this subject is intensely geometrical. only six diagrams are included in the discussion. Homology is introduced via a categorical approach in the next chapter. Singular homology on the category of topological pairs and simplicial homology on the category of simplicial pairs. The author begins the chapter with a nice intuitive discussion, but then quickly runs off to an extremely abstract definition-theorem-proof treatment of homology theory. The discussion reads like one straight out of a book on homological algebra. This approach is even more apparent in the next chapter, where homology theory is extended to general coefficient groups. The Steenrod squaring operations, which have a beautiful geometric interpretation, are instead treated in this chapter as cohomology operations. The logic used is impeccable but the real understanding gained is severely lacking. General cohomology theory is treated in the next chapter with the duality between homology and cohomology investigated via the slant product. Characteristic classes, so important in applications, are discussed using algebraic constructions via the cup product and Steenrod squares. Characteristic classes do have a nice geometric interpretation, but this is totally masked in the discussion here. The higher homotopy groups and CW complexesare discussed in Chapter 7, but again, the functorial approach used here totally obscures the underlying geometrical constructions. Obstruction theory is the subject of Chapter8, with Eilenberg-Maclane spaces leading off the discussion. The author does give some motivation in the first few paragraphs on how obstructions arise as an impediment to a lifting of a map, but an explicit, concrete example is what is needed here. The last chapter covers spectral sequences as applied to homotopy groups of spheres. More homological algebra again, and the same material could be obtained (and in more detail) in a book on that subject.
That said, if you already know the subject Spanier'sbook is an excellent reference. Even here, though, you'll need to providesome details toward the ends of the later chapters. Each chapter starts outrelatively easily and works up to a crescendo, the treatment becomingterser and more advanced. I give it four stars (5 for mathematicalquality, 3 for usefulness as a text). The first three chapters deal withcovering spaces and fibrations; the middle three with (co)homology andduality; the last three with general homotopy theory, obstruction theory,and spectral sequences. Some of Serre's classical results on finitenesstheorems for homotopy groups are presented.
That said, if you already know the subject Spanier'sbook is an excellent reference.Even here, though, you'll need to providesome details toward the ends of the later chapters.Each chapter startsout relatively easily and works up to a crescendo, the treatment becomingterser and more advanced. I give it four stars (5 for mathematicalquality, 3 for usefulness as a text).The first three chapters deal withcovering spaces and fibrations; the middle three with (co)homology andduality; the last three with general homotopy theory, obstruction theory,and spectral sequences.Some of Serre's classical results on finitenesstheorems for homotopy groups are presented. ... Read more |

2. ALGABRAIC TOPOLOGYby Edwin H. Spanier | |

Hardcover:
Pages
(1966)
Asin: B000HGKXJUCanada | United Kingdom | Germany | France | Japan | |

3. Set theory and metric spacesby Edwin Henry Spanier | |

Paperback: 82
Pages
(1955)
Asin: B0007FFI1OCanada | United Kingdom | Germany | France | Japan | |

4. Algebraic Topology '66by Edwin H. Spanier | |

Paperback:
Pages
(1966)
Asin: B000OFN8Y6Canada | United Kingdom | Germany | France | Japan | |

5. Obstruction theory, (Notas de matematica)by Edwin Henry Spanier | |

Unknown Binding: 55
Pages
(1966)
Asin: B0006BYDECCanada | United Kingdom | Germany | France | Japan | |

6. Research on duality in homotopy theory between the Mathematics Division, Air Force Office of Scientific Research, ARDC, and the University of Chicago Department ... Office of Scientific Research. TN 59-359)by Edwin Henry Spanier | |

Unknown Binding:
Pages
(1959)
Asin: B0007K90SGCanada | United Kingdom | Germany | France | Japan | |

7. Linear geometry (mathematics 112)by Edwin Henry Spanier | |

Unknown Binding:
Pages
(1961)
Asin: B0007H31CUCanada | United Kingdom | Germany | France | Japan | |

8. General topologyby Edwin Henry Spanier | |

Unknown Binding:
Pages
(1960)
Asin: B0007K792KCanada | United Kingdom | Germany | France | Japan | |

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A B C D E F G H I J K L M N O P Q R S T U V W X Y Z |

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