Homology Theory. An Introduction To Algebraic Topology (graduate Texts In Mathematics Vol. 145)

This book is designed to be an introduction to some of the basic ideas in the field of algebraic topology. In particular, it is devoted to the foundations and applications of homology theory. The only prerequisite for the student is a basic knowledge of abelian groups and point set topology. The essentials of singular homology are given in the first chapter, along with some of the most important applications. In this way the student can quickly see the importance of the material. The successive topics include attaching spaces, finite CW complexes, the Eilenberg-Steenrod axioms, cohomology products, manifolds, Poincaré duality, and fixed point theory. Throughout the book the approach is as illustrative as possible, with numerous examples and diagrams. Extremes of generality are sacrificed when they are likely to obscure the essential concepts involved. The book is intended to be easily read by students as a textbook for a course or as a source for individual study. The second edition has been substantially revised. It includes a new chapter on covering spaces in addition to illuminating new exercises.;1 Singular Homology Theory -- 2 Attaching Spaces with Maps -- 3 The Eilenberg-Steenrod Axioms -- 4 Covering Spaces -- 5 Products -- 6 Manifolds and Poincaré Duality -- 7 Fixed-Point Theory -- Appendix I -- Appendix II -- References -- Books and Historical Articles Since 1973 -- Books and Notes -- Survey and Expository Articles. This Book Is Designed To Be An Introduction To Some Of The Basic Ideas In The Field Of Algebraic Topology. In Particular, It Is Devoted To The Foundations And Applications Of Homology Theory. The Only Prerequisite For The Student Is A Basic Knowledge Of Abelian Groups And Point Set Topology. The Essentials Of Singular Homology Are Given In The First Chapter, Along With Some Of The Most Important Applications. In This Way The Student Can Quickly See The Importance Of The Material. The Successive Topics Include Attaching Spaces, Finite Cw Complexes, The Eilenberg-steenrod Axioms, Cohomology Products, Manifolds, Poincare Duality, And Fixed Point Theory. Throughout The Book, The Approach Is As Illustrative As Possible, With Numerous Examples And Diagrams. Extremes Of Generality Are Sacrificed When They Are Likely To Obscure The Essential Concepts Involved. The Book Is Intended To Be Easily Read By Students As A Textbook For A Course Or As A Source For Individual Study. This Second Edition Has Been Expanded To Include A New Chapter On Covering Spaces, As Well As Additional Illuminating Exercises. The Conceptual Approach Is Again Used To Show How Lifting Problems Give Rise To The Fundamental Group And Its Properties. Singular Homology Theory -- Attaching Spaces With Maps -- The Eilenberg-steenrod Axioms -- Covering Spaces -- Products -- Manifolds And Poincaré Duality -- Fixed-point Theory. James W. Vick. Originally Published: New York : Academic Press, C1973. (pure And Applied Mathematics ; V. 53). Includes Bibliographical References And Index. The 20 years since the publication of this book have been an era of continuing growth and development in the field of algebraic topology. New generations of young mathematicians have been trained, and classical problems have been solved, particularly through the application of geometry and knot theory. Diverse new resources for introductory coursework have appeared, but there is persistent interest in an intuitive treatment of the basic ideas. This second edition has been expanded through the addition of a chapter on covering spaces. By analysis of the lifting problem it introduces the funda­ mental group and explores its properties, including Van Kampen's Theorem and the relationship with the first homology group. It has been inserted after the third chapter since it uses some definitions and results included prior to that point. However, much of the material is directly accessible from the same background as Chapter 1, so there would be some flexibility in how these topics are integrated into a course. The Bibliography has been supplemented by the addition of selected books and historical articles that have appeared since 1973. This Book Is Designed To Be An Introduction To Some Of The Basic Ideas In The Field Of Algebraic Topology. In Particular, It Is Devoted To The Foundations And Applications Of Homology Theory. The Only Prerequisite For The Student Is A Basic Knowledge Of Abelian Groups And Point Set Topology. The Essentials Of Singular Homology Are Given In The First Chapter, Along With Some Of The Most Important Applications. In This Way The Student Can Quickly See The Importance Of The Material. The Successive Topics Include Attaching Spaces, Finite Cw Complexes, The Eilenberg-steenrod Axioms, Cohomology Products, Manifolds, Poincaré Duality, And Fixed Point Theory. Throughout The Book The Approach Is As Illustrative As Possible, With Numerous Examples And Diagrams. Extremes Of Generality Are Sacrificed When They Are Likely To Obscure The Essential Concepts Involved. The Book Is Intended To Be Easily Read By Students As A Textbook For A Course Or As A Source For Individual Study. The Second Edition Has Been Substantially Revised. It Includes A New Chapter On Covering Spaces In Addition To Illuminating New Exercises. By James W. Vick.

This book is designed to be an introduction to some of the basic ideas in the field of algebraic topology. In particular, it is devoted to the foundations and applications of homology theory. The only prerequisite for the student is a basic knowledge of abelian groups and point set topology. The essentials of singular homology are given in the first chapter, along with some of the most important applications. In this way the student can quickly see the importance of the material. The successive topics include attaching spaces, finite CW complexes, the Eilenberg-Steenrod axioms, cohomology products, manifolds, Poincaré duality, and fixed point theory. Throughout the book the approach is as illustrative as possible, with numerous examples and diagrams. Extremes of generality are sacrificed when they are likely to obscure the essential concepts involved. The book is intended to be easily read by students as a textbook for a course or as a source for individual study. The second edition has been substantially revised. It includes a new chapter on covering spaces in addition to illuminating new exercises.

This introduction to some basic ideas in algebraic topology is devoted to the foundations and applications of homology theory. After the essentials of singular homology and some important applications are given, successive topics covered include attaching spaces, finite CW complexes, cohomology products, manifolds, Poincare duality, and fixed point theory. This second edition includes a chapter on covering spaces and many new exercises. This introduction to algebraic topology concentrates on the foundations and applications of homology theory. Topics discussed include attaching spaces, finite CW complexes, the Eilenberg-Steenrod axioms, cohomology products, manifolds, Poincare duality and fixed point theory.