Algebraic Topology from a Homotopical Viewpoint

The purpose of this book is to introduce algebraic topology using the novel approach of homotopy theory, an approach with clear applications in algebraic geometry as understood by Lawson and Voevodsky. This method allows the authors to cover the material more efficiently than the more common method using homological algebra. The basic concepts of homotopy theory, such as fibrations and cofibrations, are used to construct singular homology and cohomology, as well as K-theory. Throughout the text many other fundamental concepts are introduced, including the construction of the characteristic classes of vector bundles. Although functors appear constantly throughout the text, no knowledge about category theory is expected from the reader. This book is intended for advanced undergraduates and graduate students with a basic knowledge of point set topology as well as group theory and can be used in a two semester course.
Marcelo Aguilar and Carlos Prieto are Professors at the Instituto de Matemticas, Universidad Nacional Autónoma de México, and Samuel Gitler is a member of El Colegio Nacional and professor at the Centro de Investigación y Estudios Avanzados del IPN.

This is an introductory text for a first course in algebraic topology. The authors present the beginning material in algebraic topology from a novel point of view in using a homotopy-theoretic approach. This point of view clearly has applications in algebraic geometry as understood by Lawson and Voevodsky. This carefully written book can be read by any student who knows some topology. It will be a useful place to quickly learn this novel homotoy-theoretic point of view of algebraic topology.

Function Spaces.- Connectedness and Algebraic Invariants.- Homotopy groups.- Homotopy Extension and Lifting Properties.- CW-Complexes and Homology.- Homotopy Properties of CW-Complexes.- Cohomology Groups and Related Topics.- Vector Bundles.- K-Theory.- Adams Operations and Applications.- Relations Between Cohomology and Vector Bundles.- Cohomology Theories and Brown Representability.