Homotopy Type Theory homotopy groups of spheres > history (Rev #3, changes)

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Here’s a quick reference for the state of the art on homotopy groups of spheres in HoTT. Everything listed here is also discussed on the page on Formalized Homotopy Theory.

In progress

$\pi_4(S^3)$

• Guillaume has proved that there exists an n such that π4(S3) is Z_n, and given a computational interpretation, we could run this proof and check that n is 2.$n$ such that $\pi_4(S^3)$ is $\mathbb{Z}_n$. Given a computational interpretation, we could run this proof and check that $n$ is 2.

$\pi_3(S^2)$

• Peter L. has constructed the Hopf fibration as a dependent type. Lots of people around know the construction, but I don’t know anywhere it’s written up. Here’s some Agda code with it in it.
• Guillaume’s proof that the total space of the Hopf fibration is $S^3$, together with $\pi_n(S^n)$, imply this by a long-exact-sequence argument, but this hasn’t been formalized.

At least one proof has been formalized

Freudenthal Suspension Theorem

Implies $\pi_k(S^n) = \pi_{k+1}(S^{n+1})$ whenever $k \le 2n - 2$

• Peter’s encode/decode-style proof, formalized by Dan, using a clever lemma about maps out of two n-connected types.

π_n(Sn)$\pi_n(S^n)$

• Dan and Guillaume’s encode/decode-style proof using iterated loop spaces (for single-loop presentation).
• Guillaume’s proof (for suspension definition).
• Dan’s proof from Freudenthal suspension theorem (for suspension definition).

π_k(Sn) for k < n$\pi_k(S^n)$ for $k \lt n$

• Guillaume’s proof (see the book).
• Dan’s encode/decode-style proof for pi^1(S^2) pi_1(S^2) only.
• Dan’s encode/decode-style proof for all k < n (for single-loop presentation).
• Dan’s proof from Freudenthal suspension theorem (for suspension definition).

π2(S2)$\pi_2(S^2)$

• Guillaume’s proof using the total space the Hopf fibration.
• Dan’s encode/decode-style proof.

π1(S1)$\pi_1(S^1)$

• Mike’s proof by contractibility of total space of universal cover (HoTT blog).
• Dan’s encode/decode-style proof (HoTT blog). A paper mostly about the encode/decode-style proof, but also describing the relationship between the two.
• Guillaume’s proof using the flattening lemma.