The author, a leading figure in algebraic topology, provides a modern treatment of a long established set of questions in this important research area. The book's principal objective--and main result--is the classification theorem on k-variants and boundary invariants, which supplement the classical picture of homology and homotopy groups, along with computations of types that are obtained by applying this theorem. Research mathematicians in algebraic topology will be interested in this new attempt to classify homotopy types of simply connected CW-complexes.

This book gives a general outlook on homotopy theory.

Compactness in topology and finite generation in algebra are nice properties to start with. However, the study of compact spaces leads naturally to non-compact spaces and infinitely generated chain complexes; a classical example is the theory of covering spaces. In handling non-compact spaces we must take into account the infinity behaviour of such spaces. This necessitates modifying the usual topological and algebraic cate- gories to obtain "proper" categories in which objects are equipped with a "topologized infinity" and in which morphisms are compatible with the topology at infinity. The...

The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics.

The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic.

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Lev Birbrair,...

In this book we consider deep and classical results of homotopy theory like the homological Whitehead theorem, the Hurewicz theorem, the finiteness obstruction theorem of Wall, the theorems on Whitehead torsion and simple homotopy equivalences, and we characterize axiomatically the assumptions under which such results hold. This leads to a new combinatorial foundation of homology and homotopy. Numerous explicit examples and applications in various fields of topology and algebra are given.

In this work the author extends results of rational homotopy theory to a subring of the rationale.