Advertisement

manuscripta mathematica

, Volume 152, Issue 1–2, pp 167–188 | Cite as

On the higher order exterior and interior Whitehead products

  • Marek Golasiński
  • Thiago de MeloEmail author
Article
  • 61 Downloads

Abstract

We extend the notion of the exterior Whitehead product for maps \(\alpha _i{:}\,\Sigma A_i \rightarrow X_i\) for \(i=1,\ldots ,n\), where \(\Sigma A_i\) is the reduced suspension of \(A_i\) and then, for the interior product with \(X_i=J_{m_i}(X)\), the \(m_i\)th-stage of the James construction J(X) as well. The main result stated in Theorem 4.10 generalizes (Hardie in Q J Math Oxford Ser 12(2):196–204, 1961, Theorem 1.10) and concerns to the Hopf invariant of the generalized Hopf construction. We close the paper applying Gray’s construction \(\circ \) (called the Theriault product) to a sequence \(X_1,\ldots ,X_n\) of simply connected co-H-spaces to obtain a higher Gray–Whitehead product map
$$\begin{aligned} w_n{:}\,\Sigma ^{n-2}(X_1\circ \cdots \circ X_n)\rightarrow T_1(X_1,\ldots ,X_n), \end{aligned}$$
where \(T_1(X_1,\ldots ,X_n)\) is the fat wedge of \(X_1,\ldots ,X_n\).

Mathematics Subject Classification

Primary 55Q15 Secondary 55Q25 55S15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ando, H.: On the generalized Whitehead products and the generalized Hopf invariant of a composition element. Tôhoku Math. J. 20(2), 516–553 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arkowitz, M.: Whitehead products as images of Pontrjagin products. Trans. Am. Math. Soc. 158, 453–463 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arkowitz, M.: Introduction to Homotopy Theory. Universitext. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  4. 4.
    Arnold, V.I.: Topological content of the Maxwell theorem on multipole representation of spherical functions. Topol. Methods Nonlinear Anal. 7(2), 205–217 (1996)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Baues, H.J.: Hopf invariants for reduced products of spheres. Proc. Am. Math. Soc. 59(1), 169–174 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blakers, A.L., Massey, W.S.: Products in homotopy theory. Ann. Math. 58, 295–324 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Boardman, J.M., Steer, B.: On Hopf invariants. Comment. Math. Helv. 42, 180–221 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Davis, M.W., Januszkiewicz, T.: Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J. 62, 417–451 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Golasiński, M., de Melo, T.: On the higher Whitehead product. J. Homotopy Relat. Struc. (to appear)Google Scholar
  10. 10.
    Gray, B.: On generalized Whitehead products. Trans. Am. Math. Soc. 11, 6143–6158 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grbić, J., Theriault, S.: Higher Whitehead products in toric topology. arXiv:1011.2133v2 [math.AT]
  12. 12.
    Hardie, K.A.: On a construction of EC Zeeman. J. Lond. Math. Soc. 35, 452–464 (1960)CrossRefzbMATHGoogle Scholar
  13. 13.
    Hardie, K.A.: A generalization of the Hopf construction. Q. J. Math. Oxford Ser. 12(2), 196–204 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hardie, K.A.: Higher Whitehead products. Q. J. Math. Oxford Ser. 12(2), 241–249 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Iriye, K., Kishimoto, D.: Polyhedral products for shifted complexes and higher Whitehead products. arXiv:1505.04892 [math.AT]
  16. 16.
    James, I.M.: Reduced product spaces. Ann. Math. 62, 170–197 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    James, I.M.: On the suspension sequence. Ann. Math. 65, 74–107 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Morton, H.R.: Symmetric products of the circle. Math. Proc. Cambridge Philos. Soc. 63, 349–352 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nakaoka, M., Toda, H.: On Jacobi identity for Whitehead products. J. Inst. Polytech Osaka City Univ. Ser. A 5(1), 1–13 (1954)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Porter, G.J.: Higher order Whitehead products. Topology 3, 123–135 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Porter, G.J.: The homotopy groups of wedges of suspensions. Am. J. Math. 88, 655–663 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Porter, G.J.: Higher order Whitehead products and Postnikov systems. Ill. J. Math. 11, 414–416 (1967)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Puppe, D.: Homotopiemengen und ihre induzierten Abbildungen. I. Math. Z. 69, 299–344 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rutter, J.W.: Two theorems on Whitehead products. J. Lond. Math. Soc. 43, 509–512 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rutter, J.W.: Correction to “Two theorems on Whitehead products”. J. Lond. Math. Soc. 1(2), 20 (1969)Google Scholar
  26. 26.
    Salvatore, P.: Homotopy type of Euclidean configuration spaces. Rend. Circ. Mat. Palermo 66(2 Suppl), 161–164 (2001)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Shar, A.: \(\pi _{mn-2}(S^n_{m-2})\) contains an element of order \(m\). Proc. Am. Math. Soc. 34, 303–306 (1972)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Toda, H.: Generalized Whitehead product and homotopy groups of spheres. J. Inst. Polytech. Osaka City Univ. Ser. A Math. 3, 43–82 (1952)MathSciNetGoogle Scholar
  29. 29.
    Toda, H.: Composition methods in homotopy groups of spheres. Annals of Mathematics Sudies, vol. 49, pp. 193. Princeton University Press, NJ (1962)Google Scholar
  30. 30.
    Tsuchida, K.: Generalized James product and the Hopf construction. Tôhoku Math. J. 17(4), 319–334 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Whitehead, G.W.: A generalization of the Hopf invariant. Ann. Math. 51, 192–237 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Whitehead, G.W.: Elements of Homotopy Theory. Graduate Texts in Mathematics, vol. 61. Springer, Berlin (1978)zbMATHGoogle Scholar
  33. 33.
    Wu, J.: On Maps from Loop Suspensions to Loop Spaces and the Shuffle Relations on the Cohen Groups. Memoirs of the American Mathematical Society, vol. 180, no. 851 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceUniversity of Warmia and MazuryOlsztynPoland
  2. 2.Instituto de Geociências e Ciências ExatasUNESP–Univ Estadual PaulistaRio ClaroBrazil

Personalised recommendations