Whitehead Products as Images of Pontrjagin Products

Abstract

A method is given for computing higher order Whitehead products in the homotopy groups of a space $X$. If $X$ can be embedded in an $H$-space $E$ such that the pair $(E,X)$ has sufficiently high connectivity, then we prove that a higher order Whitehead product element in the homotopy of $X$ is the homomorphic image of a Pontrjagin product in the homology of $E$. The two main applications determine a higher order Whitehead product element in (1) ${\pi _ \ast }(B{U_t})$, the homotopy groups of the classifying space of the unitary group ${U_t}$, (2) the homotopy groups of a space with two nonvanishing homotopy groups.