Foundations Of Combinatorial Topology | L. S. Pontryagin

This book represents essentially a semester course in combinatorial topology which I have given several times at the Moscow National Uni­ versity. It contains a very rigorous but concise presentation of homology theory. The formal prerequisites are merely a few simple facts about func­ tions of a real variable, matrices, and commutative groups. Actually, how­ ever, considerable mathematical maturity is required of the reader. An essential defect in the book is its complete omission of examples, which are so indispensable for clarifying the geometric content of combinatorial topology. In this sense a good complementary volume would be Sketch of the Fundamental Notions of Topology by Alexandrov and Efremovitch, in which the attention is focused on the geometric content rather than on the completeness and rigor of proofs. In spite of this shortcoming, it seems to me that the present work has certain advantages over the existing volum­ inous treatises, especially in view of its brevity. It can be used as a reference for obtaining preliminary information required for participation in a ser­ ious seminar on combinatorial topology. It is convenient in preparing for an examination in a course, since the proofs are carried out in the book with sufficient detail. For a more qualified reader, e.g., an aspiring mathema­ tician, it can also serve as a source of basic information on combinatorial

topology.

The present book makes use of a few facts concerning metric spaces

which are now ordinarily included in a course in the theory of functions of a real variable, and which can be found in the sixth chapter of Hausdorff’s Mengenlehre or in the third chapter of Alexandrov and Kolmogorov’s Theory of Functions of a Real Variable. Information concerning commuta­ tive groups may be found in the fifth chapter (see §§21 and 22) of Kurosh’s Theory of Groups.