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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Chapter I. SURVEY OF THE ELEMENTARY PROPERTIES OF TOPOLOGICAL SPACES 2
Chapter II. THE JORDAN THEOREM 39
Chapter III. SURFACES 66
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
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
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
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