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, 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


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 


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© 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

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