# More Concise Algebraic Topology: Localization, Completion, and Model Categories (Chicago Lectures in Mathematics) (Hardcover)

With firm foundations dating only from the 1950s, algebraic topology is a relatively young area of mathematics. There are very few textbooks that treat fundamental topics beyond a first course, and many topics now essential to the field are not treated in any textbook. J. Peter May’s *A Concise Course in Algebraic Topology* addresses the standard first course material, such as fundamental groups, covering spaces, the basics of homotopy theory, and homology and cohomology. In this sequel, May and his coauthor, Kathleen Ponto, cover topics that are essential for algebraic topologists and others interested in algebraic topology, but that are not treated in standard texts. They focus on the localization and completion of topological spaces, model categories, and Hopf algebras. The first half of the book sets out the basic theory of localization and completion of nilpotent spaces, using the most elementary treatment the authors know of. It makes no use of simplicial techniques or model categories, and it provides full details of other necessary preliminaries. With these topics as motivation, most of the second half of the book sets out the theory of model categories, which is the central organizing framework for homotopical algebra in general. Examples from topology and homological algebra are treated in parallel. A short last part develops the basic theory of bialgebras and Hopf algebras.

**J. P. May** is professor of mathematics at the University of Chicago; he is the author or coauthor of many papers and books, including* Simplicial Objects in Algebraic Topology *and *A Concise Course in Algebraic Topology*, both in the Chicago Lectures in Mathematics series.

**Kathleen Ponto** is assistant professor of mathematics at the University of Kentucky.

— Zentralblatt MATH

“May and Ponto have done an excellent job of assembling important results scattered throughout the mathematical literature, primarily in research articles, into a coherent, compelling whole. All researchers in algebraic topology should have at least a passing acquaintance with the material treated in this book, much of which does not appear in any of the standard texts.”

— Kathryn Hess, Ecole Polytechnique Fédérale de Lausanne