User’s Guide to Algebraic Topology

This volume arose from courses taught by the authors, and is designed for both instructional and reference use during and after a first course in algebraic topology. It is a handbook for users who want to calculate, but whose main interests are in applications using the current literature rather than in developing the theory. Typical areas of applications are differential geometry and theoretical physics. The text starts with numerous pictures to illustrate the fundamental ideas and constructions in homotopy theory that are needed in later chapters. After this, it is shown how to calculate homotopy groups, homology groups and cohomology rings of most of the major theories, exact homotopy sequences of fibrations, some important spectral sequences, and all the obstructions that can be computed from these. The approach used is to mix illustrative examples with those proofs that actually develop transferable calculational aids. Extensive appendices with notes on background material are given, along with tables.