It is well know that homotopy groups of spheres are extremely difficult to compute.
One way to compute these groups is using Postnikov approximation or Whitehead tower and Serre's spectual sequence argument. This method is explained in detail in Chapter 18 of Bott and Tu's book Differential Forms in Algebraic Topology.
But this method involves complicated constructions like loop spaces and Eilenberg-Maclane spaces which makes it hard (at least to me) to see what maps between spheres realy generate the homotopy groups.
So I want to know:
1.Is there any method to construct maps between spheres which generate the homotopy groups of spheres computed in this manner?
2.How do the explicit generators relate to the constructions in this method?
Best Answer
There is no easy and explicit way to produce generators (or even nonzero classes) of stable homotopy groups with the possible exception of the image of the J-homomorphism. That being said, the Thom-Pontryagin construction, which gives an isomorphism between stable homotopy groups and cobordism classes of framed manifolds, can be used to give convenient descriptions of some (families of) homotopy classes. In particular, any compact Lie group has a (left or right) invariant framing, and the mentioned Hopf maps $\eta$ and $\nu$ correspond to the Lie groups U(1), SU(2) as manifolds with that framing, while $\sigma$ is not realized by a Lie group. For more on this, go to E. Ossa, "Lie groups as framed manifolds", and the references given there.