Review by Choice Review
The modus operandi of algebraic topology is to associate algebraic invariants, such as groups or rings, to a topological space in such a way that equivalent spaces exhibit isomorphic invariants; here, "equivalent" may be chosen to fit the geometry of the problem. In this book, "equivalent" means homotopy equivalent, i.e., equivalent spaces may be deformed in a continuous way between each other. From this approach, the sets of homotopy classes of continuous maps between spaces form the groups or rings of invariants. Of particular importance are the homotopy groups of a space; Aguilar, Prieto (both, Universidad Nacional Autonoma de Mexico) and Gitler (Centro de Investigacion y Estudios Avanzados del IPN) show how constructions applied to a space lead to fundamental algebraic invariants by determining the homotopy groups of the newly constructed space. The Dold-Thom theorem plays a starring role as it leads to the homology groups of a space. By exploiting the kinds of constructions that mimic algebraic constructions, the strongest results in algebraic topology are obtained. The authors present an account of K-theory, characteristic classes, and generalized cohomology theories after developing the fundamental framework of algebraic topology "from a homotopical viewpoint.'' For its clarity and directness, a welcome addition to advanced mathematics collections. Upper-division undergraduates through faculty. J. McCleary Vassar College
Copyright American Library Association, used with permission.
Review by Choice Review
