We obtain upper
bounds for the stable suspension orders of the summands in Snaith’s stable
decomposition of Ω2kS2n+1, for 2k < 2n + 1, and localized at a prime p. These
finite, torsion complexes also occur as subquotients of May’s filtration of
Ω2kS2n+1, and as Thom spaces of canonical vector bundles. The results are
obtained by induction, using results of Toda on the stable orders of stunted real
projective spaces (for p = 2) and certain cofibrations of Mahowald to start
the induction and proceding by stabilizing factorizations of power maps on
Ω2kS2n+1 due to James and Selick for p = 2 and Cohen-Moore-Neisendorfer for
p > 2.
