# Homotopy Type Theory Formalized Homotopy Theory (Rev #10)

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

### $\pi_4(S^3)$

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

• On 14 February 2016 Guillaume announced a proof that $n=2$, via a proof involving the Thom construction and Gysin sequences.

### $\pi_3(S^2)$

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

### Cohomology

• To do cohomology with finite coefficients, all we need is the boring work of defining $\mathbb{Z}/p\mathbb{Z}$ 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

### Freudenthal Suspension Theorem

Implies $\pi_k(S^n) = \pi_{k+1}(S^{n+1})$ whenever $k \leq 2n - 2$

• 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 $2n$-connected.

### $\pi_2(S^2)$

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

### $\pi_1(S^1)$

• 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

### Covering spaces

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

### Blakers-Massey

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

category: homotopy theory

Revision on August 25, 2018 at 20:16:35 by Anonymous?. See the history of this page for a list of all contributions to it.