In this post, we will see the three volume set of Combinatorial Topology by P. S. Aleksandrov.

Vol. 1: Introduction. Complexes. Coverings. Dimension.

Vol. 2: The Betti Groups

Vol. 3: Homological Manifolds. The Duality Theorems. Cohomology Groups of Compacta. Continuous Mappings of Polyhedra.

# About the books

Volume 1 is a translation of the first third of P. S. Aleksandrov’s Kombinatornaya Topologiya. An appendix on the analytic geometry of Euclidean n-space is also included. The volume, complete in itself, deals with certain classical problems such as the Jordan curve theorem and the classification of closed surfaces without using the formal techniques of homology theory. The elementary but rigorous treatment of these problems, the introductory chapters on complexes and coverings and their applications to dimension theory, and the large number of examples and pictures should provide an excellent intuitive background for further study in combinatorial topology.In Chapter I the references have been expanded to include a number of standard works in English. References to these and to the books and papers cited in Chapter I of the original are listed at the end of the chapter and correspond to the numbers enclosed in brackets in the body of the text. References in the remaining chapters are enclosed in brackets, capital letters referring to books and lower case letters to papers. These refer to the bibliography at the end of the book. The bibliography includes all papers mentioned in the original edition and a few which have been added by the translator. Volume 2 of this three volume set on Combinatorial Topology covers Betti Groups and Delta-Groups comprehensively. Third volume covers Homological manifolds, the duality theorems, cohomology groups of compacta, continuous mappings of polyhedra.

The books were translated from Russian by Horace Komm was first published in 1956 and third reprint by 1969.

Credits to original uploader.

You can get the Volume 1 here.

You can get the Volume 2 here.

You can get the Volume 3 here.

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# Contents

# VOLUME 1

## PART ONE

INTRODUCTION

Chapter I. SURVEY OF THE ELEMENTARY PROPERTIES OF TOPOLOGICAL SPACES 2

Chapter II. THE JORDAN THEOREM 39

Chapter III. SURFACES 66

## PART TWO

COMPLEXES. COVERINGS. DIMESION

Chapter IV. COMPLEXES 116

Chapter V. SPERNER’S LEMMA AND ITS CORROLORIES 156

CHAPTER VI. INTRODUCTION TO DIMENSION THEORY 170

APPENDIX 1 202

BIBLIOGRAPHY 216

INDEX 218

# VOLUME 2

## PART THREE

THE BETTI GROUPS

CHAPTER VII. CHAINS. THE OPERATOR 𝝙 2

CHAPTER VIII. 𝝙-GROUPS OF COMPLEXES (LOWER BETTI OR HOMOLOGY GROUPS) 50

CHAPTER IX. THE OPERATOR 𝝙 AND THE GROUPS 𝝙^r. CANOICAL BASES. CALCULATIONS OF THE GROUPS 𝝙^r (𝕽,𝖀) and 𝝙^r (𝕽,𝖀) BY MEANS OF GROUPS 𝝙^r_0 (𝕽) 90

CHAPTER X. INVARIANCE OF THE BETTI GROUPS 125

CHAPTER XI. THE 𝛥-GROUPS OF COMPACTA 158

CHAPTER XII. THE RELATIVE CYCLES AND THEIR APPLICATIONS 178

APPENDIX 2 210

LIST OF SYMBOLS 238

INDEX 241

# VOLUME 3

## PART FOUR

HOMOLOGICAL MANIFOLDS. THE DUALITY THEOREMS. COHOMOLOGY GROUPS OF COMPACTA

CHAPTER XIII. HOMOLOGICAL MANIFOLDS (h-MANIFOLDS) 4

CHAPTER XIV. COHOMOLOGY GROUPS OF COMPACTA AND THE ALEXANDER PONTRYAGIN DUALITY 41

XV. LINKING. THE LITTLE ALEXANDER DUALITY 73

## PART FIVE

INTRODUCTION TO THE THEORY OF CONTINUOUS MAPPINGS OF POLYHEDRA

XVI. THE BROUWER THEORY OF CONTINUOUS MAPPING IN R^n AND S^n 100

XVII. FIXED POINTS OF CONTINUOUS MAPPINGS OF POLYHEDRA 128

REFERENCES 146

INDEX 147

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