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