Simplicial and Operad Methods in Algebraic Topology
About this Title
V. A. Smirnov, Moscow State Pedagogical Institute, Moscow, Russia. Translated by Professor G. L. Rybnikov
Publication: Translations of Mathematical Monographs
Publication Year: 2001; Volume 198
ISBNs: 978-0-8218-2170-1 (print); 978-1-4704-4624-6 (online)
MathSciNet review: MR1811110
MSC: Primary 55P48; Secondary 18D50, 18G30, 55P15, 55S20, 55S30, 55U35
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.
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
- Homology 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