EHP spectral sequence

In mathematics, the EHP spectral sequence is a spectral sequence used for inductively calculating the homotopy groups of spheres localized at some prime "p". It is described in more detail in and . It is related to the EHP long exact sequence of ; the name "EHP" comes from the fact that George W. Whitehead named 3 of the maps of his sequence "E" (the first letter of the German word "Einhängung" meaning "suspension"), "H" (for Heinz Hopf, as this map is the second Hopf–James invariant), and "P" (related to Whitehead products).
For formula_1 the spectral sequence uses some exact sequences associated to the fibration 
where formula_3 stands for a loop space and the (2) is localization of a topological space at the prime 2. This gives a spectral sequence with formula_4 term equal to 
and converging to formula_6 (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 to calculate the first 31 stable homotopy groups of spheres.
For arbitrary primes one uses some fibrations found by : 
where formula_9 is the formula_10-skeleton of the loop space formula_11. (For formula_1, the space formula_9 is the same as formula_14, so Toda's fibrations at formula_1 are the same as the James fibrations.)