Eilenberg-MacLane: Collected works. (English) Zbl 0613.01018

Orlando etc.: Academic Press, Inc. Harcourt Brace Jovanovich, Publishers. X, 841 p.; Cloth: $ 69.50; Paper: $ 39.50 (1986).
The present volume contains a selection of works of Samuel Eilenberg and Saunders MacLane, two of the most famous contemporary mathematicians, whose fruitful collaboration between the years 1940 and 1954 had a tremendous impact on the development of modern algebra and topology. The papers reprinted here are grouped into five ”blocks”, according to thematic, rather than chronological, criteria. Block A consists of five joint papers and one by MacLane; they contain some rudiments of homological algebra inspired by work in algebraic topology (such as the universal coefficient theorem and the functor Ext), and some concepts which later led to the development of category theory. Four other joint papers and one by MacLane are contained in block B and deal with the cohomology theory of groups, including such results as reduction theorems, and description of cohomology groups by obstructions and extensions, applications to Galois cohomology, and the relations to the algebraic theory of loops. Block C contains five joint papers by the two authors, two by Eilenberg, two by Eilenberg and J. A. Zilber, and one paper by MacLane and J. H. C. Whitehead. These papers include some of the most important results concerning the relations between homology and homotopy groups of spaces and the first approach to what are today known as the Eilenberg-MacLane spaces. The explicit calculations of the homology and cohomology groups of Eilenberg-MacLane spaces and their use in describing cohomology operations are the main subjects treated in the nine joint papers and in one by MacLane, which make up block D. The final block E, consisting of three joint papers, is devoted to such special topics as acyclic models and generic acyclicity.
Reviewer: J.Weinstein


01A75 Collected or selected works; reprintings or translations of classics
18-03 History of category theory
20-03 History of group theory
55-03 History of algebraic topology