Foundations of Combinatorial Topology (Dover Books on Mathematics) by L .S. Pontryagin (PDF)

Description

Hailed by The Mathematical Gazette as “an extremely valuable addition to the literature of algebraic topology,” this concise but rigorous introductory treatment focuses on applications to dimension theory and fixed-point theorems. The lucid text examines complexes and their Betti groups, including Euclidean space, application to dimension theory, and decomposition into components; invariance of the Betti groups, with consideration of the cone construction and barycentric subdivisions of a complex; and continuous mappings and fixed points. Proofs are presented in a complete, careful, and elegant manner.In addition to its value as a one-semester text for graduate-level courses, this volume can also be used as a reference in preparing for seminars or examinations and as a source of basic information on combinatorial topology. Although considerable mathematical maturity is required of readers, formal prerequisites are merely a few simple facts about functions of a real variable, matrices, and commutative groups.

User’s Reviews

Editorial Reviews: From the Back Cover Hailed by The Mathematical Gazette as “an extremely valuable addition to the literature of algebraic topology,” this concise but rigorous introductory treatment focuses on applications to dimension theory and fixed-point theorems. The lucid text examines complexes and their Betti groups, including Euclidean space, application to dimension theory, and decomposition into components; invariance of the Betti groups, with consideration of the cone construction and barycentric subdivisions of a complex; and continuous mappings and fixed points. Proofs are presented in a complete, careful, and elegant manner.In addition to its value as a one-semester text for graduate-level courses, this volume can also be used as a reference in preparing for seminars or examinations and as a source of basic information on combinatorial topology. Although considerable mathematical maturity is required of readers, formal prerequisites are merely a few simple facts about functions of a real variable, matrices, and commutative groups.Dover (2015) republication of the edition originally published by the Graylock Press, Rochester, New York, 1952, and reprinted by Dover Publications, Inc., in 1999.See every Dover book in print atwww.doverpublications.com About the Author L. S. Pontryagin (1908–88) was a prominent Soviet mathematician who made important contributions to many fields, including topology and differential topology.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book covers, as the title says, the foundations of combinatorial topology. What that refers to, essentially, is the topology of polyhedra and the machinery of simplicial homology groups. The first part of the book establishes the basic facts about these topological spaces and about abstract complexes as well. The second part shows the topological invariance of simplicial homology. The third features applications of the material to fixed point theory. Throughout the book there is also a recurring digression on dimension theory, which culminates in a proof of the very non-trivial fact that manifolds have the appropriate topological dimension. The only formal pre-requisites for this book are basic linear algebra and point-set topology, as covered in any advanced calculus or elementary real analysis course. The presentation is very concise and lucid throughout. I can imagine some people not liking such an extremely concise style, but the whole book is so logical that it works very well.

⭐Je n’ai pas reçu le bon produit, j’ai reçu le tome 3 d’Alexandrof “Combinatorial topology”

⭐

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