Here are some links to proofs about homotopy theory, formalized in homotopy type theory. Please add!

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.

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.

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

In progress## π4(S3)

## π3(S2)

CohomologyAt least one proof has been formalized## Whitehead's theorem

## K(G,n)

## Freudenthal Suspension Theorem

Implies π_k(Sn) = π_k+1(Sn+1) whenever k <= 2n - 2## π_n(Sn)

## π_k(Sn) for k < n

## π2(S2)

## π1(S1)

A paper mostly about the encode/decode-style proof, but also describing the relationship between the two.

## Homotopy limits

## Van Kampen

## Covering spaces

## Blakers-Massey