Kervaire invariant and Whitehead square

In this note, we try to shed some light on the relationships between the Hopf invariant, Kervaire invariant, and Whitehead square. More specifically, we prove two classical results (theorems 2.1 and 3.1) in a way that seems more transparent (at least to the author) than what is found in the literature. No new results are claimed, only a different exposition. In addition to the references cited, most of the ideas in section 3 come from discussions with Michael Hopkins, for which the author is grateful. Notation. We will use the standard notation ιn ∈ πn(S n) for the class of the identity map, and by Whitehead square we mean the Whitehead product [ιn, ιn] ∈ π2n−1(S n).