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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?

Liu Hang
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    Another reference for the Postnikov tower / Serre spectral sequence approach is Mosher and Tangora, Cohomology operations and applications in homotopy theory. – Dan Isaksen Apr 04 '11 at 18:01
  • What kind of answer would you want to question #1? An explicit formula for a function $f:S^n \to S^m$? While that's possible for the various Hopf maps, what seems more essential is the fact that they are fibrations. I think the qualitative aspects of these functions is in some sense more useful than an actual formula. Tilman's answer about the Thom-Pontrjagin construction gives such a qualitative description, by providing the framed bordism type of the preimage of a regular value under $f$. – Craig Westerland Apr 05 '11 at 12:24

4 Answers4

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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.

Tilman
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The answer partly depends on your definition of "explicit generator". The Hopf maps $\eta$, $\nu$, and $\sigma$ have explicit constructions. After that, things get messier. One way to describe the generators is with Toda brackets (see Toda, Composition methods in homotopy groups of spheres).

For example, the element often called $\epsilon$ in $\pi_8$ can be described with the bracket $\langle \eta, 2, \nu^2 \rangle$.

One word of caution regarding Toda brackets: beware of the indeterminacies.

Dan Isaksen
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  • A very late reply, but it's worth noting that Whitehead squares admit explicit constructions and generate some unstable homotopy groups (such as \pi_5(S^3)). – Mathstudent1996 Jan 13 '21 at 12:31
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Some work has been done on trying to represent homotopy classes between spheres by polynomial or rational maps. The work originated in the 1960's with papers of Reg Wood and Paul Baum:

Wood, R., Polynomial maps from spheres to spheres, Invent. Math. 5, 163-168 (1968). ZBL0204.23805.

Baum, P. F., Quadratic maps and stable homotopy groups of spheres, Ill. J. Math. 11, 586-595 (1967). ZBL0166.19102.

More recent and complete results can be found in:

Peng, Jiagui; Tang, Zizhou, Algebraic maps from spheres to spheres, Sci. China, Ser. A 42, No. 11, 1147-1154 (1999). ZBL0993.14002.

This is about as explicit as you can get, and indeed the work of Baum hints that at some stage people were thinking about doing calculations of homotopy groups by studying equivalence classes of quadratic forms. However I don't think this really caught on, and I have no idea of the relation between these algebraic constructions and the homotopy methods you describe in your question.

Mark Grant
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Not quite a solution for the OP's question, but the article Combinatorial group theory and the homotopy groups of finite complexes by Roman Mikhailov and Jie Wu, Geom. Topol. 17 (2013), no. 1, 235-272, may help.

Abdelmalek Abdesselam
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