Basic algebraic topology / Anant R. Shastri
- Shastri, Anant R.
- Boca Raton : Chapman and Hall/CRC, 
- Physical Description:
- xv, 535 pages ; 27 cm
- Machine generated contents note: 1.1.The Basic Problem -- 1.2.Fundamental Group -- 1.3.Function Spaces and Quotient Spaces -- 1.4.Relative Homotopy -- 1.5.Some Typical Constructions -- 1.6.Cofibrations -- 1.7.Fibrations -- 1.8.Categories and Functors -- 1.9.Miscellaneous Exercises to Chapter 1 -- 2.1.Basics of Convex Polytopes -- 2.2.Cell Complexes -- 2.3.Product of Cell Complexes -- 2.4.Homotopical Aspects -- 2.5.Cellular Maps -- 2.6.Abstract Simplicial Complexes -- 2.7.Geometric Realization of Simplicial Complexes -- 2.8.Barycentric Subdivision -- 2.9.Simplicial Approximation -- 2.10.Links and Stars -- 2.11.Miscellaneous Exercises to Chapter 2 -- 3.1.Basic Definitions -- 3.2.Lifting Properties -- 3.3.Relation with the Fundamental Group -- 3.4.Classification of Covering Projections -- 3.5.Group Action -- 3.6.Pushouts and Free Products -- 3.7.Seifert-van Kampen Theorem -- 3.8.Applications -- 3.9.Miscellaneous Exercises to Chapter 3 -- 4.1.Basic Homological Algebra -- 4.2.Singular Homology Groups -- 4.3.Construction of Some Other Homology Groups -- 4.4.Some Applications of Homology -- 4.5.Relation between π1 and H1 -- 4.6.All Postponed Proofs -- 4.7.Miscellaneous Exercises to Chapter 4 -- 5.1.Set Topological Aspects -- 5.2.Triangulation of Manifolds -- 5.3.Classification of Surfaces -- 5.4.Basics of Vector Bundles -- 5.5.Miscellaneous Exercises to Chapter 5 -- 6.1.Method of Acyclic Models -- 6.2.Homology with Coefficients: The Tor Functor -- 6.3.Kunneth Formula -- 6.4.Miscellaneous Exercises to Chapter 6 -- 7.1.Cochain Complexes -- 7.2.Universal Coefficient Theorem for Cohomology -- 7.3.Products in Cohomology -- 7.4.Some Computations -- 7.5.Cohomology Operations; Steenrod Squares -- 8.1.Orientability -- 8.2.Duality Theorems -- 8.3.Some Applications -- 8.4.de Rham Cohomology -- 8.5.Miscellaneous Exercises to Chapter 8 -- 9.1.Sheaves -- 9.2.Injective Sheaves and Resolutions -- 9.3.Cohomology of Sheaves -- 9.4.Cech Cohomology -- 9.5.Miscellaneous Exercises to Chapter 9 -- 10.1.H-spaces and H'-spaces -- 10.2.Higher Homotopy Groups -- 10.3.Change of Base Point -- 10.4.The Hurewicz Isomorphism -- 10.5.Obstruction Theory -- 10.6.Homotopy Extension and Classification -- 10.7.Eilenberg-Mac Lane Spaces -- 10.8.Moore-Postnikov Decomposition -- 10.9.Computation with Lie Groups and Their Quotients -- 10.10.Homology with Local Coefficients -- 10.11.Miscellaneous Exercises to Chapter 10 -- 11.1.Generalities about Fibrations -- 11.2.Thom Isomorphism Theorem -- 11.3.Fibrations over Suspensions -- 11.4.Cohomology of Classical Groups -- 11.5.Miscellaneous Exercises to Chapter 11 -- 12.1.Orientation and Euler Class -- 12.2.Construction of Steifel-Whitney Classes and Chern Classes -- 12.3.Fundamental Properties -- 12.4.Splitting Principle and Uniqueness -- 12.5.Complex Bundles and Pontrjagin Classes -- 12.6.Miscellaneous Exercises to Chapter 12 -- 13.1.Warm-up -- 13.2.Exact Couples -- 13.3.Algebra of Spectral Sequences -- 13.4.Leray-Serre Spectral Sequence -- 13.5.Some Immediate Applications -- 13.6.Transgression -- 13.7.Cohomology Spectral Sequences -- 13.8.Serre Classes -- 13.9.Homotopy Groups of Spheres.
- "Thoroughly classroom-tested, this self-contained text teaches algebraic topology to students at the MSc and PhD levels, taking them all the way to becoming algebraic topologists. Requiring basic training in point set topology, linear algebra, and group theory, the book includes historical remarks to make the subject more meaningful to students. Also suitable for researchers, it provides references for further reading, presents full proofs of all results, and includes numerous exercises"--
"PREFACE This book is intended for a 2-semester first course in algebraic topology, though I would recommend not to try to cover the whole thing in two semesters. A glance through the contents page will tell the reader that the selection of topics is quite standard whereas the sequencing of them may not be so. The material in the first five chapters are very basic and quite enough for a semester course. A teacher can afford to be a little choosy in selecting exactly which sections (s)he may want to teach. There is more freedom in choice of materials to be taught from latter chapters. It goes without saying that these materials demand much higher mathematical maturity than the first five chapters. Also, this is where some knowledge of differential manifolds helps to understand the material better. The book can be adopted as a text for M.Sc./B.Tech./M.Tech./Ph.D. students. We assume that the readers of this book have gone through a semester course each in real analysis, and point-set-topology and some basic algebra. It is desirable that they also had a course in differential topology or concurrently study such a course but that is necessary only at a few sections. There are exercises at the end of many sections and at the end of first five chapters. Most of these exercises are part of the main material and working through them is an essential part of learning. However, it is not necessary that a student gets the right answers before proceeding further. Indeed, it is not a good idea to get stuck with a problem for too long--keep going further and come back to them later. There is a hint/solution manual for them at the end of the book for some selected exercises, especially for those which are being used in a later section, so as to make"--
- 9781466562431 (hardback)
- Bibliography Note:
- Includes bibliographical references (pages 525-529) and index.
- Source of Acquisition:
- Purchased with funds from the James and Joyce Gettys Libraries Endowment in the Math Library and in the School of Information Sciences and Technology; 2014.
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