TITLE: How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there?
QUESTION [23 upvotes]: Offhand I can think of two ways in classical homotopy theory:

Show that $\pi_k(S^n)=0$ for $k\lt n$ by deforming a map $S^k\to S^n$ to be non-surjective, then contracting it away from a point not in its image.  Now use the Hurewicz theorem to show $\pi_n(S^n) = H_n(S^n) = \mathbb{Z}$, which is easy to calculate with cellular homology.
Use the Freudenthal suspension theorem to induct up from $\pi_1(S^1)=\mathbb{Z}$, which you can prove using (say) the universal covering space $\mathbb{R}\to S^1$.

What other ways are there to prove $\pi_n(S^n)=\mathbb{Z}$?

REPLY [3 votes]: This is the first theorem proven in Whitehead's book ''Elements of homotopy theory'', section I.3. The proof is by a simplicial argument, which is similar to what I believe is Hopf's original argument (if the degree is $0$, take preimages of regular values of a map and deform to reduce the number of preimages).