For the spectral sequence uses some exact sequences associated to the fibration (James 1957)
,
where stands for a loop space and the (2) is localization of a topological space at the prime 2. This gives a spectral sequence with term equal to
and converging to (stable homotopy groups of spheres localized at 2). The spectral sequence has the advantage that the input is previously calculated homotopy groups. It was used by Oda (1977) to calculate the first 31 stable homotopy groups of spheres.
For arbitrary primes one uses some fibrations found by Toda (1962):
where is the -skeleton of the loop space . (For , the space is the same as , so Toda's fibrations at are the same as the James fibrations.)
Oda, Nobuyuki (1977), "On the 2-components of the unstable homotopy groups of spheres, I–II", Proc. Japan Acad. Ser. A Math. Sci., 53 (6): 202–218, doi:10.3792/pjaa.53.202