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Author: Holger Kammeyer
Publisher: Springer Nature
ISBN: 3030983137
Category : Mathematics
Languages : en
Pages : 186
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Book Description
This textbook provides a succinct introduction to algebraic topology. It follows a modern categorical approach from the beginning and gives ample motivation throughout so that students will find this an ideal first encounter to the field. Topics are treated in a self-contained manner, making this a convenient resource for instructors searching for a comprehensive overview of the area. It begins with an outline of category theory, establishing the concepts of functors, natural transformations, adjunction, limits, and colimits. As a first application, van Kampen's theorem is proven in the groupoid version. Following this, an excursion to cofibrations and homotopy pushouts yields an alternative formulation of the theorem that puts the computation of fundamental groups of attaching spaces on firm ground. Simplicial homology is then defined, motivating the Eilenberg-Steenrod axioms, and the simplicial approximation theorem is proven. After verifying the axioms for singular homology, various versions of the Mayer-Vietoris sequence are derived and it is shown that homotopy classes of self-maps of spheres are classified by degree.The final chapter discusses cellular homology of CW complexes, culminating in the uniqueness theorem for ordinary homology. Introduction to Algebraic Topology is suitable for a single-semester graduate course on algebraic topology. It can also be used for self-study, with numerous examples, exercises, and motivating remarks included.
Author: Holger Kammeyer
Publisher: Springer Nature
ISBN: 3030983137
Category : Mathematics
Languages : en
Pages : 186
Get Book Here
Book Description
This textbook provides a succinct introduction to algebraic topology. It follows a modern categorical approach from the beginning and gives ample motivation throughout so that students will find this an ideal first encounter to the field. Topics are treated in a self-contained manner, making this a convenient resource for instructors searching for a comprehensive overview of the area. It begins with an outline of category theory, establishing the concepts of functors, natural transformations, adjunction, limits, and colimits. As a first application, van Kampen's theorem is proven in the groupoid version. Following this, an excursion to cofibrations and homotopy pushouts yields an alternative formulation of the theorem that puts the computation of fundamental groups of attaching spaces on firm ground. Simplicial homology is then defined, motivating the Eilenberg-Steenrod axioms, and the simplicial approximation theorem is proven. After verifying the axioms for singular homology, various versions of the Mayer-Vietoris sequence are derived and it is shown that homotopy classes of self-maps of spheres are classified by degree.The final chapter discusses cellular homology of CW complexes, culminating in the uniqueness theorem for ordinary homology. Introduction to Algebraic Topology is suitable for a single-semester graduate course on algebraic topology. It can also be used for self-study, with numerous examples, exercises, and motivating remarks included.
Author: J. P. May
Publisher: University of Chicago Press
ISBN: 9780226511832
Category : Mathematics
Languages : en
Pages : 262
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Book Description
Algebraic topology is a basic part of modern mathematics, and some knowledge of this area is indispensable for any advanced work relating to geometry, including topology itself, differential geometry, algebraic geometry, and Lie groups. This book provides a detailed treatment of algebraic topology both for teachers of the subject and for advanced graduate students in mathematics either specializing in this area or continuing on to other fields. J. Peter May's approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. The final four chapters provide sketches of substantial areas of algebraic topology that are normally omitted from introductory texts, and the book concludes with a list of suggested readings for those interested in delving further into the field.
Author: James W. Vick
Publisher: Springer Science & Business Media
ISBN: 1461208815
Category : Mathematics
Languages : en
Pages : 258
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Book Description
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.
Author: Allen Hatcher
Publisher: Cambridge University Press
ISBN: 9780521795401
Category : Mathematics
Languages : en
Pages : 572
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Book Description
An introductory textbook suitable for use in a course or for self-study, featuring broad coverage of the subject and a readable exposition, with many examples and exercises.
Author: Andrew H. Wallace
Publisher: Courier Corporation
ISBN: 0486152952
Category : Mathematics
Languages : en
Pages : 212
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Book Description
This self-contained treatment begins with three chapters on the basics of point-set topology, after which it proceeds to homology groups and continuous mapping, barycentric subdivision, and simplicial complexes. 1961 edition.
Author: Rafael Ayala
Publisher: ALPHA SCIENCE INTERNATIONAL LIMITED
ISBN: 1783322446
Category : Mathematics
Languages : en
Pages : 386
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Book Description
ALGEBRAIC TOPOLOGY: An Introduction starts with the combinatorial definition of simplicial (co) homology and its main properties (including duality for homology manifolds). Then the geometrical facet of (co) homology via bordism theory is sketched and it is shown that the corresponding theory for pseudomanifolds coincides with the homology obtained from the singular chain complex. The classical applications of (co) homology theory are included. Degree and fixed-point theory are presented with their extensions to infinite dimensional spaces. The book also contains a geometric approach to the Hurewicz theorem relating homology and homotopy. The last chapter exploits the algebraic invariants introduced in the book to give in detail the homotopical classification of the three-dimensional lens spaces. Each chapter concludes with a generous list of exercises and problems; many of them contain hints for their solution. Some groups of problems introduce a topic not included in the basic core of the book.
Author: C. R. F. Maunder
Publisher: Courier Corporation
ISBN: 9780486691312
Category : Mathematics
Languages : en
Pages : 414
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Book Description
Based on lectures to advanced undergraduate and first-year graduate students, this is a thorough, sophisticated, and modern treatment of elementary algebraic topology, essentially from a homotopy theoretic viewpoint. Author C.R.F. Maunder provides examples and exercises; and notes and references at the end of each chapter trace the historical development of the subject.
Author: William S. Massey
Publisher: Springer
ISBN: 1493990632
Category : Mathematics
Languages : en
Pages : 448
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Book Description
This textbook is intended for a course in algebraic topology at the beginning graduate level. The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory. These topics are developed systematically, avoiding all unnecessary definitions, terminology, and technical machinery. The text consists of material from the first five chapters of the author's earlier book, Algebraic Topology; an Introduction (GTM 56) together with almost all of his book, Singular Homology Theory (GTM 70). The material from the two earlier books has been substantially revised, corrected, and brought up to date.
Author: V. A. Vasilʹev
Publisher: American Mathematical Soc.
ISBN: 0821821628
Category : Mathematics
Languages : en
Pages : 165
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Book Description
This English translation of a Russian book presents the basic notions of differential and algebraic topology, which are indispensable for specialists and useful for research mathematicians and theoretical physicists. In particular, ideas and results are introduced related to manifolds, cell spaces, coverings and fibrations, homotopy groups, homology and cohomology, intersection index, etc. The author notes, "The lecture note origins of the book left a significant imprint on itsstyle. It contains very few detailed proofs: I tried to give as many illustrations as possible and to show what really occurs in topology, not always explaining why it occurs." He concludes, "As a rule, only those proofs (or sketches of proofs) that are interesting per se and have importantgeneralizations are presented."
Author: Glen E. Bredon
Publisher: Springer Science & Business Media
ISBN: 1475768486
Category : Mathematics
Languages : en
Pages : 571
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Book Description
This book offers an introductory course in algebraic topology. Starting with general topology, it discusses differentiable manifolds, cohomology, products and duality, the fundamental group, homology theory, and homotopy theory. From the reviews: "An interesting and original graduate text in topology and geometry...a good lecturer can use this text to create a fine course....A beginning graduate student can use this text to learn a great deal of mathematics."—-MATHEMATICAL REVIEWS
