Advertisement

Products in Homotopy Theory

  • Mahima Ranjan Adhikari
Chapter

Abstract

This chapter continues to study homotopy theory through different products defined between homotopy groups such as the Whitehead product introduced by J.H.C. Whitehead in 1941, the Samelson product introduced by H.

Keywords

Homotopy Class Jacobi Identity Homotopy Group Mixed Product Algebraic Topology 
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, 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.: Whitehead products as images of Pontrjagin products. Trans. Am. Math. Soc. 158, 453–463 (1971)MathSciNetCrossRefMATHGoogle Scholar
  4. Arkowitz, M., Curjel, C.R.: Some properties of the exotic multiplications on the three-sphere. Q. J. Math. Oxf. Ser. 20(2), 171–176 (1969)MathSciNetCrossRefMATHGoogle Scholar
  5. Blakers, A.L., Massy, W.S.: Products in homotopy theory. Ann. Math. 58, 295–324 (1953)MathSciNetCrossRefMATHGoogle Scholar
  6. Dieudonné, J.: A History of Algebraic and Differential Topology, pp. 1900–1960. Modern Birkhäuser, Boston (1989)Google Scholar
  7. Gilbert, W.J.: Homotopical nilpotence of the seven sphere. Proc. Am. Math. Soc. 32, 621–622 (1972)MathSciNetCrossRefMATHGoogle Scholar
  8. Gray, B.: Homotopy Theory. An Introduction to Algebraic Topology. Acamedic Press, New York (1975)MATHGoogle Scholar
  9. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)MATHGoogle Scholar
  10. Hilton, P.J.: Homotopy Theory and Duality. Nelson, London (1965)Google Scholar
  11. Hilton, P.J., Whitehead, J.H.C.: Notes on the Whitehead product. Ann. Math. 58(2), 429–442 (1953)MathSciNetCrossRefMATHGoogle Scholar
  12. Hilton, P.J.: Note on the Jacobi identity for Whitehead products. Proc. Camb. Philos. Soc. 57, 180–182 (1961)MathSciNetCrossRefMATHGoogle Scholar
  13. Hu, S.T.: Homotopy Theory. Academic Press, New York (1959)MATHGoogle Scholar
  14. James, I.M.: Products between homotopy groups. Compositio Mathematica 23, 329–45 (1971)MathSciNetMATHGoogle Scholar
  15. James, I.M., Thomas, E.: Which Lie groups are homotopy abelian? Proc. Nat. Acc. Sc. USA 25, 131–140 (1959)MathSciNetMATHGoogle Scholar
  16. Massey, W.S., Uehara, H.: The Jacobi Identity for Whitehead Products. Princeton University Press, Princeton (1957)MATHGoogle Scholar
  17. Maunder, C.R.F.: Algebraic Topology. Van Nostrand Reinhold Company, London (1980)MATHGoogle Scholar
  18. May, J.P.: A Concise Course in Algebraic Topology. University of Chicago Press, Chicago (1999)MATHGoogle Scholar
  19. McCarty, G.S.: Products between homotopy groups and \(J\)-morphism. Q. J. Math. Oxf. 15(2), 362–370 (1964)MathSciNetCrossRefMATHGoogle Scholar
  20. Miyazaki, H.: On realizations of some Whitehead products. Tohoku Math. J. 12, 1–30 (1960)MathSciNetCrossRefMATHGoogle Scholar
  21. Nakaoka, M., Toda, H.: On Jacobi identity for Whitehead products. J. Inst. Polytech, Osaka City Univ. Ser A 5, 1–13 (1954)Google Scholar
  22. Samelson, H.: A connection between the Whitehead product and Pontragin product. Am. J. Math 75, 744–752 (1953)MathSciNetCrossRefMATHGoogle Scholar
  23. Stephen, J.S.: A Samelson product and homotopy associativity. Proc. Am. Math. Soc 70(2), 189–195 (1978)MathSciNetCrossRefMATHGoogle Scholar
  24. Spanier, E.: Algebraic Topology. McGraw-Hill, New York (1966)MATHGoogle Scholar
  25. Toda, H.: Generalized Whitehead products and homotopy groups of spheres. J. Inst. Polytech. Osaka City Univ. Ser. A,3, 43-48 (1953)Google Scholar
  26. Whitehead, G.W.: On products in homotopy groups. Ann. Math. 47(2), 460–475 (1944)MathSciNetMATHGoogle Scholar
  27. Whitehead, G.W.: Elements of Homotopy Theory. Springer, Heidelberg (1978)Google Scholar
  28. Whitehead, J.H.C.: On adding relations to homotopy groups. Ann. of Math. 42(2), 409–428 (1941)Google Scholar

Copyright information

© Springer India 2016

Authors and Affiliations

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

Personalised recommendations