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

  • 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.

At least one proof has been formalized

Freudenthal Suspension Theorem

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.


π_k(Sn) for k < n


  • 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.