More elementary proof that $\pi_n(S^n) \cong \mathbb{Z}$

Question

The proof I know that $\pi_n(S^n) \cong \mathbb{Z}$ is based on the Hurewicz theorem (which implies that $\pi_n(S^n) \cong H_n(S^n)$). I'm looking for a more elementary argument - preferably something which doesn't require any homology theory. Any ideas?

Answer 1

Historically this was a consequence of the Hopf degree theorem. Given $f: S^n \rightarrow S^n$ with induced map $f_*: H_n(S^n) \rightarrow H_n(S^n)$ we have $f\sim 0$ iff $f_*=0$. This is proved by taking some kind of approximation to $f$, either differentiable, in which case a proof can be found in Guillman and Pollack, Differentiable Topology. Or a simplicial approximation, for which a proof can be found in the early pages of Whiteheads homotopy theory. Also chapter 16 of Dugundji has a proof of the Hopf theorem. In particular this gives that the map $\deg :\pi_n(S^n) \rightarrow \mathbb{Z}$ is an isomorphism. Of course this is again just a special case of Hurewicz.

Answer 2

There's the Freudenthal suspension theorem:

For $i<2n-1$, the suspension map $\pi_i(S^n)\to\pi_{i+1}(S^{n+1})$ is an isomorphism, and for $i=2n-1$, it is a surjection.

See Hatcher, Corollary 4.25. Consider the sequence of suspensions $\pi_1(S^1)\to\pi_2(S^2)\to\cdots$. By the Freudenthal suspension theorem, $\pi_1(S^1)\to\pi_2(S^2)$ is surjective. $\pi_1(S^1)\cong \mathbf{Z}$, so $\pi_n(S^n)$ is a finite or infinite cyclic group. It's infinite since, avoiding homology, the long exact sequence of homotopy groups of the Hopf bundle $S^1\to S^3\to S^2$ shows that $\pi_1(S^1)\cong\pi_2(S^2)$. Hatcher concludes by:

The degree map $\pi_n(S^n)\to\mathbf{Z}$ is an isomorphism since the map $z\mapsto z^k$ of $S^1$ has degree $k$, as do its iterated suspensions by Proposition 2.33 (which states that $\mathrm{deg} Sf=\mathrm{deg}f$, where $Sf:S^{n+1}\to S^{n+1}$ is the suspension of $f:S^n\to S^n$).

This isn't very elementary, but avoids homology.

Answer 3

The book partially titled Nonabelian Algebraic Topology, (with downloadable pdf), published in 2011 by the EMS, gives a full account of a different approach to algebraic topology avoiding setting up a homology theory, or using simplicial approximation.

Not only does this work obtain the Brouwer Degree Theorem, and the Relative Hurewicz Theorem, (!), but it also obtains results on second relative homotopy groups, and nonabelian computation of homotopy $2$-types, not available by other means.

I gave a talk, available here, at the IHP in Paris in June to a workshop on Homotopy Type Theory; this gives the intuitive background, emphasising the use of higher homotopy groupoids of structured spaces, and the notion of multiple compositions. This avoids the use of free abelian groups, by generalising to higher dimensions the fundamental groupoid on a set of base points.

The book was written with the intention of making the results of a 40 year research journey available in one place, for ease of understanding and evaluation.

The proofs are not "elementary", but assume only some knowledge of CW-complexes, and of category theory.

The Freudenthal Suspension Theorem is another story, see the ncatlab on the Blakers-Massey theorem.