# Homotopy Type Theory Formalized Homotopy Theory (Rev #2)

Maybe we could figure out a way not to duplicate stuff between this page and homotopy groups of spheres?

Cast of characters so far: Jeremy Avigad, Guillaume Brunerie, Favonia, Eric Finster, Chris Kapulkin, Dan Licata, Peter Lumsdaine, Mike Shulman.

## In progress

### π4(S3)

• Guillaume proved that there is some n such that π4(S3) is Z_n. With a computational interpretation, we could run this proof and check that it’s 2.

### π3(S2)

• Peter’s construction of 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.

### Cohomology

• Prove that K(G,n) is a spectrum (Eric?).
• To do cohomology with finite coefficients, all we need is the boring work of defining Z/pZ as an explicit group.
• Calculate some cohomology groups.

## 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 2 n-connected types. This is the “95%” version, which shows that the relevant map is an equivalence on truncations.
• The full “100%” version, formalized by Dan, which shows that the relevant map is 2n-connected.

### π2(S2)

• Guillaume’s proof.
• Dan’s encode/decode-style proof.

### π1(S1)

• 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.