Homotopy Type Theory
Formalized Homotopy Theory (Rev #7)

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

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, Evan Cavallo

In progress

π 4(S 3)\pi_4(S^3)

  • Guillaume proved that there is some nn such that π 4(S 3)\pi_4(S^3) is /n\mathbb{Z}/n\mathbb{Z}. With a computational interpretation, we could run this proof and check that n=2n=2.

π 3(S 2)\pi_3(S^2)

  • 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 S 3S^3, together with π n(S n)\pi_n(S^n), imply this by a long-exact-sequence argument, but this hasn’t been formalized.


  • To do cohomology with finite coefficients, all we need is the boring work of defining /p\mathbb{Z}/p\mathbb{Z} as an explicit group.
  • Calculate some more cohomology groups.
  • Compute the loop space of this construction and use it to define spectra.

At least one proof has been formalized

Whitehead’s theorem



Freudenthal Suspension Theorem

Implies π k(S n)=π k+1(S n+1)\pi_k(S^n) = \pi_{k+1}(S^{n+1}) whenever k2n2k \leq 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 2n2n-connected.

π n(S n)\pi_n(S^n)

π k(S n)\pi_k(S^n) for k<nk \lt n

π 2(S 2)\pi_2(S^2)

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

π 1(S 1)\pi_1(S^1)

  • 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.

Homotopy limits

  • Jeremy Avigad, Chris Kapulkin and Peter Lumsdaine arXiv Coq code

Van Kampen

Covering spaces

  • Favonia’s proofs (link in flux due to library rewrite).


  • Favonia/Peter/Guillaume/Dan’s formalization of Peter/Eric/Dan’s proof (link in flux due to library rewrite).

Revision on February 14, 2016 at 17:55:06 by David Roberts. See the history of this page for a list of all contributions to it.