Basic Algebraic Topology and its Applications

1. Front Matter

   Pages i-xxix

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2. Prerequisite Concepts and Notations

   • Mahima Ranjan Adhikari

   Pages 1-44

3. Homotopy Theory: Elementary Basic Concepts

   • Mahima Ranjan Adhikari

   Pages 45-106

4. The Fundamental Groups

   • Mahima Ranjan Adhikari

   Pages 107-145

5. Covering Spaces

   • Mahima Ranjan Adhikari

   Pages 147-196

6. Fiber Bundles, Vector Bundles and K-Theory

   • Mahima Ranjan Adhikari

   Pages 197-247

7. Geometry of Simplicial Complexes and Fundamental Groups of Polyhedra

   • Mahima Ranjan Adhikari

   Pages 249-271

8. Higher Homotopy Groups

   • Mahima Ranjan Adhikari

   Pages 273-304

9. CW-Complexes and Homotopy

   • Mahima Ranjan Adhikari

   Pages 305-327

10. Products in Homotopy Theory

    • Mahima Ranjan Adhikari

    Pages 329-346

11. Homology and Cohomology Theories

    • Mahima Ranjan Adhikari

    Pages 347-406

12. Eilenberg–MacLane Spaces

    • Mahima Ranjan Adhikari

    Pages 407-417

13. Eilenberg–Steenrod Axioms for Homology and Cohomology Theories

    • Mahima Ranjan Adhikari

    Pages 419-431

14. Consequences of the Eilenberg–Steenrod Axioms

    • Mahima Ranjan Adhikari

    Pages 433-443

15. Applications

    • Mahima Ranjan Adhikari

    Pages 445-473

16. Spectral Homology and Cohomology Theories

    • Mahima Ranjan Adhikari

    Pages 475-509

17. Obstruction Theory

    • Mahima Ranjan Adhikari

    Pages 511-531

18. More Relations Between Homology and Homotopy

    • Mahima Ranjan Adhikari

    Pages 533-545

19. A Brief History of Algebraic Topology

    • Mahima Ranjan Adhikari

    Pages 547-568

20. Back Matter

    Pages 569-615

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This book provides an accessible introduction to algebraic topology, a field 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 offers 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.

• Homotopy and cohomotopy group
• Vector bundle
• Homology and cohomology group
• Eilenberg-Steenrod axioms
• Spectrum of spaces
• Generalized cohomology
• Cohomology operations
• Hopf invariant and Knot group

“Adhikari’s work is an excellent resource for any individual seeking to learn more about algebraic topology. By no means will this text feel like an introduction to algebraic topology, but it does offer much for both beginners and experts. … the text will be a valuable reference on the bookshelf of any reader with an interest in algebraic topology. Summing Up: Recommended. Upper-division undergraduates and above; researchers and faculty.” (A. Misseldine, Choice, Vol. 54 (9), May, 2017)

“I am pretty enthusiastic about this book. … it shows very good taste on the author’s part as far as what he’s chosen to do and how he’s chosen to do it. … Wow! What a nice book. I’m glad I have a copy.” (Michael Berg, MAA Reviews, maa.org, February, 2017)

“This is a comprehensive textbook on algebraic topology. … accessible to students of all levels of mathematics, so suitable for anyone wanting and needing to learn about algebraic topology. It can also offer a valuable resource for advanced students with a specialized knowledge in other areas who want to pursue their interest in this area. … further readings are provided at the end of each of them, which also enables students to study the subject discussed therein in more depth.” (Haruo Minami, zbMATH 1354.55001, 2017)

• Bioinformatics, Information Technol, The Institute for Mathematics, Kolkata, India

  Mahima Ranjan Adhikari

Mahima Ranjan Adhikari, PhD, is a former professor of Pure Mathematics at the University of Calcutta. His main interest lies in algebra and topology. He has published a number of papers in several international journals including the Proceedings of American Mathematical Society and five textbooks. Eleven students have already been awarded the PhD degree under his supervision. He is a member of the American Mathematical Society and serves on the editorial board of several journals and research monographs. He was the president of the mathematical science section of the 95th Indian Science Congress, 2008.


He has visited several institutions in India, USA, UK, Japan, France, Greece, Sweden, Switzerland, Italy and many other countries on invitation. While visiting E.T.H., Zurich, Switzerland, in 2003, he made an academic interaction with Professor B Eckmann and P J Hilton. He is currently the president of the Institute for Mathematics, Bioinformatics, Information Technology and Computer Science (IMBIC). He is also the principal investigator of an ongoing project funded by the Government of India.


The present book is written based on author’s teaching experience of 50 years. He is the joint author ( with Avishek Adhikari) of another Springer book “Basic Modern Algebra with Applications”, 2014.

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