[Math] How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there

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Offhand I can think of two ways in classical homotopy theory:

  1. 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.

  2. 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 $\pi_n(S^n)=\mathbb{Z}$?

Best Answer

$\pi_n(S^n)=[S^n,S^n]=\lbrace$cobordism classes of framed 0-submanifolds$\rbrace$ by the Pontrjagin-Thom construction. These are collections of points (with sign) which add up to give the degree of the maps, so this set is precisely $\mathbb{Z}$.

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