Of course it isn't really that hard – nowhere near as hard as $\pi_k(S^n)$ for $k>n$, for instance. The hardness that I'm referring to is based on the observation that apparently nobody knows how to do the calculation within the homotopy category of topological spaces. Approaches that I'm aware of include:
-Homology theory (the Hurewicz theorem)
-Degree theory
-The divergence theorem
and each of these reduces to a calculation within some other category (PL or Diff). My question is: is there something "wrong" with hTop that precludes a computation of $\pi_n(S^n)$ within that category?
Certainly if my assumption that such a proof does not exist is wrong, I would be very interested to know it.
I have been thinking and reading further about this question for the past couple days, and I wanted to summarize some of the main points in the answers and questions:
- Some techniques – e.g. the Freudenthal Suspension Theorem via the James construction or the Hurewicz Theorem with singular homology – might actually lead to proofs without any approximation arguments. But so far I'm not sure we quite have it: the proof of the Freudenthal Suspension Theorem uses the fact that $J(X)$ is homotopy equivalent to the loop space of the suspension of $X$, but the only proofs I can find of this fact use a CW structure on $X$, and similarly for the proofs of the Hurewicz theorem. Can these results be proved for the sphere without PL or smooth approximation?
- Perhaps this question is entirely wrong-headed: the techniques of PL and smooth approximation are very well adapted to homotopy theory, so why try to replace them with a language which may end up adding more complications with little additional insight? Fair enough. But the goal of this question is not to disparage or seek alternatives to existing techniques, it is to understand exactly what role they play in the theory. The statement "The identity map $S^n \to S^n$ is homotopically nontrivial and freely generates $\pi_n(S^n)$" makes no mention of CW complexes or smooth structures, yet apparently the statement is difficult or impossible to prove without that sort of language (except in the case $n=1$!) To seek an understanding of this observation is different from lamenting it.
Best Answer
I suppose that the proof that $\pi_1(S^1) \cong \mathbb{Z}$ using covering spaces is homotopy-theoretic.
The Freudenthal Suspension Theorem (via the James construction) tells us that $\Sigma: \pi_n(S^n)\to \pi_{n+1}(S^{n+1})$ is an isomorphism for $n \geq 2$ and surjective for $n =1$.
Since $S^1$ is an H-space, the suspension map $\sigma: S^1\to \Omega\Sigma S^1$ has a retraction $r: \Omega\Sigma S^1\to S^1$. Therefore $\Sigma = \sigma_*: \pi_1(S^1)\to \pi_{2}(S^{2})$ is injective (in addition to being surjection).
Now Freudenthal completes the calculation.
Note that we don't just get an abstract isomorphism, we get that these groups are generated by $[\mathrm{id}_{S^n}]$.
EDIT: Regarding getting the James Construction homotopically: the paper
derives the relevant properties using the Cube Theorems (which are about the mixing of homotopy pushouts and homotopy pullbakcs) of an earlier paper of Mather's.