Algebraic Topology--homotopy and HomologyThe author has attempted an ambitious and most commendable project. He assumes only a modest knowledge of algebraic topology on the part of the reader to start with, and he leads the reader systematically to the point at which he can begin to tackle problems in the current areas of research centered around generalized homology theories and their applications. After an account of classical homotopy theory, the author turns to homology and cohomology theories, first treating them axiomatically and then constructing them using spectra. These ideas are illustrated via a thorough development of the three main examples of ordinary homology, K-theory and bordisms. Next, the author takes up the study of products in homology and cohomology and the related questions of orientability and duality. The remainder of the book is devoted to more sophisticated techniques and methods currently in use such as characteristic classes, cohomology operations, and the Adams spectral sequence, all of which are developed in the context of generalized homology theories. This book is, all in all, a very admirable work and a valuable addition to the literature and its value is not diminished by the somewhat minor flaws mentioned. -- S.Y. Husseini. |
Contents
Some Facts from General Topology | 1 |
Categories Functors and Natural Transformations | 6 |
Homotopy Sets and Groups | 11 |
Copyright | |
20 other sections not shown
Other editions - View all
Common terms and phrases
A₁ A₂ abelian group Adams spectral sequence algebra B₁ B₂ base point BO(n C₁ cell chain complex Chapter cobordism coefficient cofibre cofinal cofunctor cohomology commutative diagram complex construct Corollary CW-complex define Definition deformation retract denote dual e₁ element En+1 epimorphism exact sequence fibration finite follows functor G₁ given H₁ hence homology theory homomorphism homotopy groups Hopf Hurewicz inclusion induces inverse isomorphism Lemma lim¹ manifolds map f microbundle monomials monomorphism morphism n-connected natural equivalence natural transformation pair principal G-bundle Proof Proposition R-module ring spectrum S-duality satisfies the wedge set of transition Sq¹ subcomplex subspace Suppose surjective Theorem Thom class topology transition functions U₁ unique vector bundle wedge axiom X,xo X₁ Y,yo Y₁ Z₂