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Obstruction Theory

  • Mahima Ranjan Adhikari
Chapter

Abstract

This chapter studies a theory known as “Obstruction Theory” by applying cohomology theory to encounter two basic problems in algebraic topology such as extension and lifting problems. Obvious examples are the homotopy extension and homotopy lifting problems. The homotopy classifications of continuous maps together with the study of extension and lifting problems, play a central role in algebraic topology.

Keywords

Cohomology Class Homotopy Class Algebraic Topology Extension Problem Homotopy Classification 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Adams, J.F.: On the non-existence of elements of Hopf invariant one. Ann. Math. 72(2), 20–104 (1960)MathSciNetCrossRefMATHGoogle Scholar
  2. Adams, J.F.: Algebraic Topology: A Student’s Guide. Cambridge University Press, Cambridge (1972)CrossRefMATHGoogle Scholar
  3. Arkowitz, M.: Introduction to Homotopy Theory. Springer, New York (2011)CrossRefMATHGoogle Scholar
  4. Davis, J.F., Kirk, P.: Lecture Notes in Algebraic Topology. Indiana University, Bloomington (2001). http://www.ams.org/bookstore-getitem/item=GSM-35 CrossRefMATHGoogle Scholar
  5. Dieudonné, J.: A History of Algebraic and Differential Topology, 1900–1960. Modern Birkhäuser Classics. Birkhäuser, Basel (1989)MATHGoogle Scholar
  6. Dodson, C.T.J., Parkar, P.E.: User’s Guide to Algebraic Topology. Kluwer Academic Publishers, Dordrecht (1997)CrossRefGoogle Scholar
  7. Dold, A.: Lectures on Algebraic Topology. Springer, New York (1972)CrossRefMATHGoogle Scholar
  8. Eilenberg, S., Steenrod, N.: Foundations of Algebraic Topology. Princeton University Press, Princeton (1952)Google Scholar
  9. Gray, B.: Homotopy Theory: An Introduction to Algebraic Topology. Academic, New York (1975)MATHGoogle Scholar
  10. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)Google Scholar
  11. Hilton, P.J.: An Introduction to Homotopy Theory. Cambridge University Press, Cambridge (1983)Google Scholar
  12. Hilton, P.J., Wylie, S.: Homology Theory. Cambridge University Press, Cambridge (1960)CrossRefMATHGoogle Scholar
  13. Hu, S.T.: Homotopy Theory. Academic Press, New York (1959)Google Scholar
  14. Massey, W.S.: A Basic Course in Algebraic Topology. Springer, New York (1991)MATHGoogle Scholar
  15. Maunder, C.R.F.: Algebraic Topology. Van Nostrand Reinhold Company, London (1970)MATHGoogle Scholar
  16. Mayer, J.: Algebraic Topology. Prentice-Hall, New Jersy (1972)Google Scholar
  17. Olum, P.: Obstructions to extensions and homotopies. Ann. Math. 25, 1–25 (1950)MathSciNetCrossRefMATHGoogle Scholar
  18. Samelson, H.: Groups and spaces of loops. Comment. Math. Helv. 28, 278–286 (1954)MathSciNetCrossRefMATHGoogle Scholar
  19. Spanier, E.: Algebraic Topology. McGraw-Hill, New York (1966)MATHGoogle Scholar
  20. Steenrod, N.E.: The Topology of Fibre Bundles. Princeton University Press (1951)Google Scholar
  21. Steenrod, N.E.: Chohomology operations and obstructions to extending continuous functions. Adv. Math. 8, 371–416 (1972)MathSciNetCrossRefMATHGoogle Scholar
  22. Switzer, R.M.: Algebraic Topology-Homotopy and Homology. Springer, Berlin (1975)CrossRefMATHGoogle Scholar
  23. Whitehead, G.W.: On mappings into group like spaces. Comment. Math. Helv. 28, 320–328 (1954)MathSciNetCrossRefGoogle Scholar
  24. Whitehead, G.W.: Elements of Homotopy Theory. Springer, New York (1978)Google Scholar

Copyright information

© Springer India 2016

Authors and Affiliations

  1. 1.Institute for Mathematics, Bioinformatics, Information Technology and Computer Science (IMBIC)KolkataIndia

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