Basic Algebraic Topology and its Applications by Mahima Ranjan Adhikari
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About this book :-
Basic Algebraic Topology and its Applications written by
Mahima Ranjan Adhikari .
This book provides an accessible introduction to algebraic topology, a ?eld at the intersection of topology, geometry and algebra, together with its applications. Moreover, it covers several related topics that are in fact important in the overall scheme of algebraic topology. Comprising eighteen chapters and two appendices, the book integrates various concepts of algebraic topology, supported by examples, exercises, applications and historical notes. Primarily intended as a textbook, the book o?ers a valuable resource for undergraduate, postgraduate and advanced mathematics students alike. Focusing more on the geometric than on algebraic aspects of the subject, as well as its natural development, the book conveys the basic language of modern algebraic topology by exploring homotopy, homology and cohomology theories, and examines a variety of spaces: spheres, projective spaces, classical groups and their quotient spaces, function spaces, polyhedra, topological groups, Lie groups and cell complexes, etc. The book studies a variety of maps, which are continuous functions between spaces. It also reveals the importance of algebraic topology in contemporary mathematics, theoretical physics, computer science, chemistry, economics, and the biological and medical sciences, and encourages students to engage in further study. Algebraic topology is one of the most important creations in mathematics which uses algebraic tools to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism (though usually classify up to homotopy equivalence). The most important of these invariants are homotopy groups, homology groups, and cohomology groups (rings). The main purpose of this book is to give an accessible presentation to the readers of the basic materials of algebraic topology through a study of homotopy, homology, and cohomology theories. Moreover, it covers a lot of topics for advanced students who are interested in some applications of the materials they have been taught. Several basic concepts of algebraic topology, and many of their successful applications in other areas of mathematics and also beyond mathematics with surprising results have been given. The essence of this method is a transformation of the geometric problem to an algebraic one which offers a better chance for solution by using standard algebraic methods.
(Mahima Ranjan Adhikari)
Book Detail :-
Title: Basic Algebraic Topology and its Applications
Edition:
Author(s): Mahima Ranjan Adhikari
Publisher: Springer
Series:
Year: 2016
Pages: 628
Type: PDF
Language: English
ISBN: 8132228413,9788132228417
Country: India
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About Author :-
Avishek Adhikari , Department of Pure Mathematics, University of Calcutta, Kolkata, West Bengal, India.
Mahima Ranjan Adhikari , IMBIC, Kolkata, West Bengal, India,
Yogendra Prasad Chaubey , Department of Mathematics and Statistics, Concordia University, Montreal, QC, Canada
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Book Contents :-
Basic Algebraic Topology and its Applications written by
Mahima Ranjan Adhikari
cover the following topics.
1. Prerequisite Concepts and Notations
2. Homotopy Theory: Elementary Basic Concepts
3. The Fundamental Groups
4. Covering Spaces
5. Fiber Bundles, Vector Bundles and K-Theory
6. Geometry of Simplicial Complexes and Fundamental Groups of Polyhedra
7. Higher Homotopy Groups
8. CW-Complexes and Homotopy
9. Products in Homotopy Theory
10. Homology and Cohomology Theories
11. Eilenberg–MacLane Spaces
12. Eilenberg–Steenrod Axioms for Homology and Cohomology Theories
13. Consequences of the Eilenberg–Steenrod Axioms
14. Applications
15. Spectral Homology and Cohomology Theories
16. Obstruction Theory
17. More Relations Between Homology and Homotopy
18. A Brief History of Algebraic Topology
Appendix A: Topological Groups and Lie Groups
Appendix B: Categories, Functors and Natural Transformations
List of Symbols
Author Index
Subject Index
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