But after all, $\pi_n(X,*)$ is "merely" the set of morphisms in the homotopy category of pointed spaces from the $n$-sphere to $X$: $\pi_n(X,*) = \hom_{\mathsf{hTop}_*}((S^n,*), (X, *))$. This has always seemed somewhat "biased" to me; spheres are given a preponderant role in the definition.
Spheres emerge naturally out of the relationship between homotopy theory and higher category theory. One example of this is that you might object that the study of $\pi_1$ somehow arbitrarily picks out $S^1$, but in fact you can define $\pi_1$ without mentioning $S^1$ at all, using the theory of covering spaces or locally constant sheaves. One definition is the following: $\pi_1(X, x)$ is the automorphism group of the functor given by taking the stalk of a locally constant sheaf on $X$ at $x$. This is a functor on the homotopy category of reasonable pointed spaces and the circle naturally emerges as the object representing this functor.
Similarly you can describe the fundamental $n$-groupoid $\Pi_n(X)$ of $X$ (which knows about the first $n$ homotopy groups) without mentioning paths at all, using a theory of "higher" locally constant sheaves (locally constant sheaves of $(n-1)$-groupoids, rather than sets).
If you believe one way or another that fundamental $n$-groupoids are reasonable things to study (inductively: if you believe that homotopies are reasonable, then you believe that homotopies between homotopies are reasonable, etc.; this means you care about the interval $[0, 1]$ but it doesn't yet obligate you to care about spheres), then the higher homotopy groups naturally emerge by repeatedly taking automorphisms:
- $\pi_1(X, x)$ is the automorphism group of $x$ as an object in the fundamental groupoid $\Pi_1(X)$,
- $\pi_2(X, x)$ is the automorphism group of the identity path $x \to x$ as an morphism in the fundamental $2$-groupoid $\Pi_2(X)$,
etc. None of this explicitly requires talking about spheres, but again, these are all very natural functors on pointed homotopy types and spheres represent them.
Said another way, among homotopy types, the $n$-sphere $S^n$ has a universal property: it is the free homotopy type on an automorphism of an automorphism of... of a point. These aren't just arbitrary spaces we picked because we like them; they're fundamental to homotopy theory in the same way that $\mathbb{Z}$ is fundamental to group theory.
All homotopy groups that can be computed in HoTT coincide with the corresponding classical homotopy groups. This is because HoTT has an interpretation in the Quillen model category of simplicial sets (constructed in a metatheory of ZFC plus some inaccessible cardinals, although probably a much weaker theory suffices), constructed by Voevodsky, which is also Quillen equivalent to classical homotopy theory. Thus, any calculation in HoTT can be interpreted into simplicial sets and then transferred across that equivalence to classical homotopy theory.
A much more interesting (and open!) question is whether every computation of a classical homotopy group is also provable in HoTT. This is interesting and nontrivial because the really interesting thing about HoTT (from my point of view) is that it also has lots of other models, namely in all higher toposes. Thus, this question is tantamount to asking whether the homotopy groups of spheres (say) are the same in all higher toposes.
(Technical fine print: we know by an extrinsic argument that the homotopy groups of spheres and other finite complexes are the same in all "Grothendieck" higher toposes, but there are also other models, and reasoning about any particular class of models can't answer the question of what can be proven internally in HoTT.)
Best Answer
At least at $p=2$, Toda's calculations do not go nearly that far up. There are later papers that go a little further, but his ``Composition methods in homotopy groups of spheres'' goes up to $n=19$. It is misleading to think of $S^2$ as particularly simple. There is an old theorem (Serre?) that if $X$ is any simply connected finite CW complex that is not contractible, such as $S^2$, then for each prime $p$ there are infinitely many $n$ such that $\pi_n(X)$ has $p$-torsion.