An Illustrated Introduction to Topology and Homotopy

Author: Sasho Kalajdzievski

Publisher: CRC Press

ISBN: 1482220814

Category: Mathematics

Page: 485

View: 5787

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An Illustrated Introduction to Topology and Homotopy explores the beauty of topology and homotopy theory in a direct and engaging manner while illustrating the power of the theory through many, often surprising, applications. This self-contained book takes a visual and rigorous approach that incorporates both extensive illustrations and full proofs. The first part of the text covers basic topology, ranging from metric spaces and the axioms of topology through subspaces, product spaces, connectedness, compactness, and separation axioms to Urysohn’s lemma, Tietze’s theorems, and Stone-Čech compactification. Focusing on homotopy, the second part starts with the notions of ambient isotopy, homotopy, and the fundamental group. The book then covers basic combinatorial group theory, the Seifert-van Kampen theorem, knots, and low-dimensional manifolds. The last three chapters discuss the theory of covering spaces, the Borsuk-Ulam theorem, and applications in group theory, including various subgroup theorems. Requiring only some familiarity with group theory, the text includes a large number of figures as well as various examples that show how the theory can be applied. Each section starts with brief historical notes that trace the growth of the subject and ends with a set of exercises.
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Introduction to Homotopy Theory

Author: Martin Arkowitz

Publisher: Springer Science & Business Media

ISBN: 9781441973290

Category: Mathematics

Page: 344

View: 3535

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This is a book in pure mathematics dealing with homotopy theory, one of the main branches of algebraic topology. The principal topics are as follows: Basic Homotopy; H-spaces and co-H-spaces; fibrations and cofibrations; exact sequences of homotopy sets, actions, and coactions; homotopy pushouts and pullbacks; classical theorems, including those of Serre, Hurewicz, Blakers-Massey, and Whitehead; homotopy Sets; homotopy and homology decompositions of spaces and maps; and obstruction theory. The underlying theme of the entire book is the Eckmann-Hilton duality theory. The book can be used as a text for the second semester of an advanced ungraduate or graduate algebraic topology course.
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Elementary Topology

Author: Michael C. Gemignani

Publisher: Courier Corporation

ISBN: 9780486665221

Category: Mathematics

Page: 270

View: 8087

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Topology is one of the most rapidly expanding areas of mathematical thought: while its roots are in geometry and analysis, topology now serves as a powerful tool in almost every sphere of mathematical study. This book is intended as a first text in topology, accessible to readers with at least three semesters of a calculus and analytic geometry sequence. In addition to superb coverage of the fundamentals of metric spaces, topologies, convergence, compactness, connectedness, homotopy theory, and other essentials, Elementary Topology gives added perspective as the author demonstrates how abstract topological notions developed from classical mathematics. For this second edition, numerous exercises have been added as well as a section dealing with paracompactness and complete regularity. The Appendix on infinite products has been extended to include the general Tychonoff theorem; a proof of the Tychonoff theorem which does not depend on the theory of convergence has also been added in Chapter 7.
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An Introduction to Algebraic Topology

Author: Joseph Rotman

Publisher: Springer Science & Business Media

ISBN: 9780387966786

Category: Mathematics

Page: 438

View: 4571

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A clear exposition, with exercises, of the basic ideas of algebraic topology. Suitable for a two-semester course at the beginning graduate level, it assumes a knowledge of point set topology and basic algebra. Although categories and functors are introduced early in the text, excessive generality is avoided, and the author explains the geometric or analytic origins of abstract concepts as they are introduced.
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Introduction to Topology and Geometry

Author: Saul Stahl,Catherine Stenson

Publisher: John Wiley & Sons

ISBN: 1118546148

Category: Mathematics

Page: 536

View: 7949

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An easily accessible introduction to over threecenturies of innovations in geometry Praise for the First Edition “. . . a welcome alternative to compartmentalizedtreatments bound to the old thinking. This clearly written,well-illustrated book supplies sufficient background to beself-contained.” —CHOICE This fully revised new edition offers the most comprehensivecoverage of modern geometry currently available at an introductorylevel. The book strikes a welcome balance between academic rigorand accessibility, providing a complete and cohesive picture of thescience with an unparalleled range of topics. Illustrating modern mathematical topics, Introduction toTopology and Geometry, Second Edition discusses introductorytopology, algebraic topology, knot theory, the geometry ofsurfaces, Riemann geometries, fundamental groups, and differentialgeometry, which opens the doors to a wealth of applications. Withits logical, yet flexible, organization, the SecondEdition: • Explores historical notes interspersed throughout theexposition to provide readers with a feel for how the mathematicaldisciplines and theorems came into being • Provides exercises ranging from routine to challenging,allowing readers at varying levels of study to master the conceptsand methods • Bridges seemingly disparate topics by creating thoughtfuland logical connections • Contains coverage on the elements of polytope theory, whichacquaints readers with an exposition of modern theory Introduction to Topology and Geometry, Second Edition is anexcellent introductory text for topology and geometry courses atthe upper-undergraduate level. In addition, the book serves as anideal reference for professionals interested in gaining a deeperunderstanding of the topic.
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A First Course in Algebraic Topology

Author: Czes Kosniowski

Publisher: CUP Archive

ISBN: 9780521298643

Category: Mathematics

Page: 269

View: 5780

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This self-contained introduction to algebraic topology is suitable for a number of topology courses. It consists of about one quarter 'general topology' (without its usual pathologies) and three quarters 'algebraic topology' (centred around the fundamental group, a readily grasped topic which gives a good idea of what algebraic topology is). The book has emerged from courses given at the University of Newcastle-upon-Tyne to senior undergraduates and beginning postgraduates. It has been written at a level which will enable the reader to use it for self-study as well as a course book. The approach is leisurely and a geometric flavour is evident throughout. The many illustrations and over 350 exercises will prove invaluable as a teaching aid. This account will be welcomed by advanced students of pure mathematics at colleges and universities.
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Cubical Homotopy Theory

Author: Brian A. Munson,Ismar Volić

Publisher: Cambridge University Press

ISBN: 1107030250

Category: Mathematics

Page: 625

View: 3270

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A modern, example-driven introduction to cubical diagrams and related topics such as homotopy limits and cosimplicial spaces.
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