We know that the homotopy groups $\pi_i(S^n)=0$ for $i<n$ and $\pi_i(S^n)\simeq\mathbb{Z}$ for $i=n$. I can understand that. This might be a dumb question as I am new to homotopy; I am not sure what is $\pi_i(S^n)$ for $i>n$?

Is it undefined or is it $0$? I am not sure how to approach it and how to think about it? Is there any useful analogy or some ways to visualise or understand it?

I couldn't find any useful theorems or properties to answer this doubt. Could anyone please give some light?

I really appreciate the helps. Many thanks!

  • 3
    $\begingroup$ It is neither undefined nor always zero. Homotopy groups are always defined. But the homotopy groups of spheres are neither known nor well-understood, and understanding these is a famous open problem in algebraic topology. $\endgroup$ – user98602 Oct 30 '16 at 8:34
  • $\begingroup$ There is even a wikipedia article on the subject, with examples, explanations and tables en.wikipedia.org/wiki/Homotopy_groups_of_spheres $\endgroup$ – lisyarus Oct 30 '16 at 8:35
  • $\begingroup$ For fixed $n$, infinitely many of the groups $\pi_i(S^n)$ contain a subgroup isomorphic to $\Bbb Z_2$. $\endgroup$ – iwriteonbananas Oct 30 '16 at 8:52
  • $\begingroup$ @iwriteonbananas How about $n\leq 1$? $\endgroup$ – Danu Oct 30 '16 at 14:57
  • $\begingroup$ Right, I was assuming $n\geq 2$. Thanks @Danu. $\endgroup$ – iwriteonbananas Oct 31 '16 at 12:51

They are well-defined: the homotopy groups $\pi_n(X)$ are well-defined for any (pointed) topological space $X$, including the spheres. They are also not zero in general, for example $\pi_3(S^2) = \mathbb{Z}$ with a generator given by the Hopf fibration.

The truth is, "nobody knows": there is no general formula giving you the homotopy groups of spheres. The Wikipedia article is a good source of information about this.

There are plenty of ways to compute a specified homotopy group (using spectral sequences and whatnot), there are general results (e.g. infinitely many groups $\pi_k(S^n)$ contain a copy of $\mathbb{Z}/2\mathbb{Z}$ for a fixed $n \ge 2$ by a result of Serre, we know the homotopy groups over $\mathbb{Q}$, all the groups are finite except for $\pi_n(S^n)$ and $\pi_{4n-1}(S^{2n})$...), there are stabilization results (by the Freudenthal suspension theorem, $\pi_{n+k}(S^n)$ becomes constant as $n \to \infty$), etc. Tons of people are working on this problem, but (to my knowledge) nobody can give you a formula saying "the homotopy groups of spheres are this".


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.