Symmetricity of the Whitehead element

摘要

We study the symmetricity of the Whitehead element w_n∈π_2np-3(S~2n-1) for an odd prime p. It is shown that w_n considered as a map S~2-3→S~2n-1 factors through the p-fold covering map σ:S~2np-3→ only when n is a power of p, and that w_pi actually factors through σ if 0≤i≤4. This is some of an odd prime version of the results of Ranadll and Lin for the projectivity of the Whitehad product (ι_2n-1,ι_2n-1)∈π_n-2(S~2n-1).