Algebraic Topology
"Algebraic Topology" by Spanier
1994 | ISBN: 0387944265 | Pages: 273 | English | DJVU | 13 MB
1994 | ISBN: 0387944265 | Pages: 273 | English | DJVU | 13 MB
This book surveys the fundamental ideas of algebraic topology. The first part covers the fundamental group, its definition and application in the study of covering spaces. The second part turns to homology theory including cohomology, cup products, cohomology operations and topological manifolds. The final part is devoted to Homotropy theory, including basic facts about homotropy groups and applications to obstruction theory.
Reader's review:
This book is terrific as a reference for those who already know the subject, but if you teach algebraic topology it would be dangerous to use it as a graduate text (unless you're willing to supplement it extensively). The basic problem is that Spanier does not teach students how to compute effectively because his abstract, high-powered algebraic approach obscures the underlying geometry, which is not developed at all. Here I'd recommend the books by Munkres, or Greenberg; even the old-fashioned treatment of Lefschetz, with its explicit and rather cumbersome treatment of cohomology, could serve as an antidote to Spanier. Somewhere, the student has to acquire a good intuitive feeling for the geometry underlying the subject (the same can be said of algebraic geometry – here earlier work (e.g., of the Italian school, Weil's old book on intersection theory, …) should not be neglected entirely in favor of Grothendieck et al., for something essential is lost)
That said, if you already know the subject Spanier's book is an excellent reference. Even here, though, you'll need to provide some details toward the ends of the later chapters. Each chapter starts out relatively easily and works up to a crescendo, the treatment becoming terser and more advanced.
I give it four stars (5 for mathematical quality, 3 for usefulness as a text). The first three chapters deal with covering spaces and fibrations; the middle three with (co)homology and duality; the last three with general homotopy theory, obstruction theory, and spectral sequences. Some of Serre's classical results on finiteness theorems for homotopy groups are presented.