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

π4(S3)

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.

π3(S2)

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.