Title: Algebraic topology : homotopy and homology /
Authors: Switzer, Robert M., 1940-
Publisher: Springer-Verlag,
Year: 1975
Subjects: Algebraic topology.; Homotopy theory.; Homology theory.; Topologie algébrique.; Homotopie.; Homologie.; Algebraic topology; Homology theory; Homotopy theory; Algebraische Topologie; Homologie; Homologietheorie; Homotopie
Abstract: The 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 -- Categories, functors and natural transformations -- Homotopy sets and groups -- Properties of the homotopy groups -- Fibrations -- CW-complexes -- Homotopy properties of CW-complexes -- Homology and cohomology theories -- Spectra -- Representation theorems -- Ordinary homology theory -- Vector bundles and K-theory -- Manifolds and bordism -- Products -- Orientation and duality -- Spectral sequences -- Characteristic classes -- Cohomology operations and homology cooperations -- The Steenrod algebra and its dual -- The Adams spectral sequence and the [italic]e-invariant -- Calculation of the cobordism groups.