Does any one have a reference to a proof that the Hopf invariant classifies the homotopy classes of maps from $\mathbb{S}^3$ to $\mathbb{S}^2$.


It is quite standard to find a proof that the Hopf invariant is a homotopy invariant. However, after searching MathSciNet for all the books covering the Hopf invariant, I could not find any proofs more recent than Hopf's original 1931 paper written in German. The best that I have found are some statements without any proof or reference.



The Hopf invariant is an isomorphism for $\pi_3 (S^2)$
As Robert Bryant and Nicholas Kuhn have indicated, this is standard stuff that is in many textbooks. One thing that might be tripping you up is that the original definition of the Hopf invariant is very geometric (involving linking numbers), while most modern treatments use alternative definitions. If you're interested in doing things in a more geometric way, see my notes "Homotopy groups of spheres and low-dimensional topology", which are available on my webpage [here](http://www.nd.edu/~andyp/notes/).


