Samuel Eilenberg

The need for an axiomatic treatment of homology and cohomology theory has long been felt by topologists. Professors Eilenberg and Steenrod present here for the first time an axiomatization of the complete transition from topology to algebra. Originally published in 1952. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

The need for an axiomatic treatment of homology and cohomology theory has long been felt by topologists. Professors Eilenberg and Steenrod present here for the first time an axiomatization of the complete transition from topology to algebra. Originally published in 1952. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Linear Algebra and Projective Geometry: Pure and Applied Mathematics, Volume 2, written by Reinhold Baer, is a comprehensive textbook that explores the relationship between linear algebra and projective geometry. The book is divided into two parts, with the first part focusing on linear algebra and the second part focusing on projective geometry. In the first part, the author covers topics such as vector spaces, linear transformations, determinants, eigenvalues, and eigenvectors. The author also discusses the theory of matrices, including matrix algebra, matrix inverses, and the computation of determinants. Additionally, the author provides numerous examples and exercises to help readers solidify their understanding of linear algebra concepts. The second part of the book delves into projective geometry, which is the study of geometric properties that are invariant under projection. The author discusses topics such as projective spaces, projective transformations, duality, and conics. The author also explores the relationship between projective geometry and linear algebra, demonstrating how linear algebra can be used to solve problems in projective geometry. Overall, Linear Algebra and Projective Geometry: Pure and Applied Mathematics, Volume 2 is a valuable resource for students and researchers in mathematics, physics, and engineering. The book provides a thorough treatment of both linear algebra and projective geometry, with numerous examples and exercises to help readers develop a deep understanding of these important topics. This scarce antiquarian book is a facsimile reprint of the old original and may contain some imperfections such as library marks and notations. Because we believe this work is culturally important, we have made it available as part of our commitment for protecting, preserving, and promoting the world's literature in affordable, high quality, modern editions, that are true to their original work.

When this book was written, methods of algebraic topology had caused revolutions in the world of pure algebra. To clarify the advances that had been made, Cartan and Eilenberg tried to unify the fields and to construct the framework of a fully fledged theory. The invasion of algebra had occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. This book presents a single homology (and also cohomology) theory that embodies all three; a large number of results is thus established in a general framework. Subsequently, each of the three theories is singled out by a suitable specialization, and its specific properties are studied. The starting point is the notion of a module over a ring. The primary operations are the tensor product of two modules and the groups of all homomorphisms of one module into another. From these, "higher order" derived of operations are obtained, which enjoy all the properties usually attributed to homology theories. This leads in a natural way to the study of "functors" and of their "derived functors." This mathematical masterpiece will appeal to all mathematicians working in algebraic topology.