Translations of Mathematical Monographs 2001; 235 pp; hardcover Volume: 198 ISBN10: 0821821709 ISBN13: 9780821821701 List Price: US$98 Member Price: US$78.40 Order Code: MMONO/198
 In recent years, for solving problems of algebraic topology and, in particular, difficult problems of homotopy theory, algebraic structures more complicated than just a topological monoid, an algebra, a coalgebra, etc., have been used more and more often. A convenient language for describing various structures arising naturally on topological spaces and on their cohomology and homotopy groups is the language of operads and algebras over an operad. This language was proposed by J. P. May in the 1970s to describe the structures on various loop spaces. This book presents a detailed study of the concept of an operad in the categories of topological spaces and of chain complexes. The notions of an algebra and a coalgebra over an operad are introduced, and their properties are investigated. The algebraic structure of the singular chain complex of a topological space is explained, and it is shown how the problem of homotopy classification of topological spaces can be solved using this structure. For algebras and coalgebras over operads, standard constructions are defined, particularly the bar and cobar constructions. Operad methods are applied to computing the homology of iterated loop spaces, investigating the algebraic structure of generalized cohomology theories, describing cohomology of groups and algebras, computing differential in the Adams spectral sequence for the homotopy groups of the spheres, and some other problems. Readership Graduate students and research mathematicians working in algebraic topology. Table of Contents  Operads in the category of topological spaces
 Simplicial objects and homotopy theory
 Algebraic structures on chain complexes
 \(A_\infty\)structures on chain complexes
 Operads and algebras over operads
 Homoloty of iterated loop spaces
 Homotopy theories and \(E_\infty\)structures
 Operad methods in cobordism theory
 Description of the cohomology of groups and algebras
 Homology operations and differentials in the Adams spectral sequence
 Bibliography
 Index
