homotopy groups of spheres (Rev #2)

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.

- 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.

- 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.

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.

- 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).

- Guillaume’s proof (see the book).
- Dan’s encode/decode-style proof for 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).

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

Revision on March 3, 2014 at 13:34:13 by Mike Shulman. See the history of this page for a list of all contributions to it.