• 2-primary component of homotopy groups
• EHP sequence
• Localization
• Unstable homotopy groups of spheres

The aim of this thesis is to construct the EHP sequence, that is a powerful tool to
calculate homotopy groups of spheres.
In order to do so we introduce a general theory of localization of spaces (due to Bousfield
and Dror Farjoun). In particular we focus on localization of simply connected spaces
at a subset of primes. Then we show that there is a fibration of spaces localized at 2
(due to James) Sn Σ→ΩSn+1 h→ΩS2n+1.
The long exact homotopy sequence associated to this fibration is the EHP sequence.
Using the above fibration we prove a theorem due to James which gives an upper
bound for the 2-primary exponent of homotopy groups of spheres; it states that 4n an-
nihilates the 2-primary component of πk(S2n+1). At the end we use the EHP sequence
to compute some homotopy groups of spheres localized at 2.