# What is $\pi_i(S^n)$ for $i>n$?

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!

• 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. – user98602 Oct 30 '16 at 8:34
• There is even a wikipedia article on the subject, with examples, explanations and tables en.wikipedia.org/wiki/Homotopy_groups_of_spheres – lisyarus Oct 30 '16 at 8:35
• For fixed $n$, infinitely many of the groups $\pi_i(S^n)$ contain a subgroup isomorphic to $\Bbb Z_2$. – iwriteonbananas Oct 30 '16 at 8:52
• @iwriteonbananas How about $n\leq 1$? – Danu Oct 30 '16 at 14:57
• Right, I was assuming $n\geq 2$. Thanks @Danu. – 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.
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".