In this post, we will see the book Foundations Of Combinatorial Topology by L. S. Pontryagin.

About the book

This book represents essentially a semester course in combinatorial topology which I have given several times at the Moscow National University. It contains a very rigorous but concise presentation of homology theory. The formal prerequisites are merely a few simple facts about functions 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 voluminous treatises, especially in view of its brevity. It can be used as a reference for obtaining preliminary information required for participation in a serious 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 mathematician, 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 commutative groups may be found in the fifth chapter (see §21 and §22) of Kurosh’s Theory of Groups.

The book was translated from Russian by F. Bagemihl, H. Komm and W. Seidel. The book was published in 1952.

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You can get the book here.

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Contents

Preface vii
Introduction ix
Notation xi

CHAPTER I COMPLEXES AND THEIR BETTI GROUPS

§ 1. Euclidean Space. I
§ 2. Simplex. Complex. Polyhedron 9
§ 3. Application to Dimension Theory. 16
§ 4. The Betti Groups. 23
§ 5. Decomposition into Components. The Zero-dimensional Betti Group. 26
§ 6. The Betti Numbers. The Euler-Poincaré Formula. 30

CHAPTER II THE INVARIANCE OF THE BETTI GROUPS

§ 7. Simplicial Mappings and Approximations. 36
§ 8. The Cone Construction. 43
§ 9. Barycentric Subdivision of a Complex. 48
§ 10. A Lemma on the Covering of a Simplex, and its Application. 53
§ 11. The Invariance of the Betti Groups under Barycentric Sub-division. 58
§12. The Invariance of the Betti Groups. 61

CHAPTER III CONTINUOUS MAPPINGS AND FIXED POINTS

§13. Homotopic Mappings. 69
§14. The Cylinder Construction. 72
§15. Homology Invariants of Continuous Mappings. 79
§16. The Existence Theorem for Fixed Points. 84

List of Definitions. 94
List of Theorems. 95

Basic Literature on Combinatorial Topology. 96