HomotopyGroupsOfSpheres

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 S3, together with π_n(Sn), imply this by a long-exact-sequence argument, but this hasn’t been formalized.

Implies π_k(Sn) = π_k+1(Sn+1) whenever k <= 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

- 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.

Created on April 19, 2018 at 17:16:58
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Univalent foundations special year 2012