Formalized Homotopy Theory (Rev #6, changes)

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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.~~ Shulman, Evan Cavallo

- 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 $n=2$.

- 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.

~~Prove~~To~~that~~do cohomology with finite coefficients, all we need is the boring work of defining$\mathrm{K\mathbb{Z}}(/\mathrm{Gp},\mathbb{Z}n)$~~K(G,n)~~\mathbb{Z}/p\mathbb{Z}~~is~~as~~a~~an~~spectrum~~explicit~~(Eric?).~~group.~~To~~Calculate~~do~~some more cohomology~~with~~groups.~~finite~~~~coefficients,~~~~all~~~~we~~~~need~~~~is~~~~the~~~~boring~~~~work~~~~of~~~~defining~~~~$\mathbb{Z}/p\mathbb{Z}$~~~~as an explicit group.~~~~Calculate~~Compute~~some~~the~~cohomology~~loop~~groups.~~space ofthis construction and use it to define spectra.

- Dan’s proof for n-types.

- Dan’s construction.
- $K(G,n)$ is a spectrum, formalized

- Deriving cohomology theories from spectra
- Mayer-Vietoris sequence, with exposition
- Cohomology of the n-torus, which could be easily extended to any finite product of spheres.

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.

- Dan and Guillaume’s encode/decode-style proof using iterated loop spaces (for single-loop presentation).
- Guillaume’s proof (for suspension definition).
- Dan’s proof from Freudenthal suspension theorem (for suspension definition).

- Guillaume’s proof (see the book).
- Dan’s encode/decode-style proof for $\pi_1(S^2)$ only.
- Dan’s encode/decode-style proof for all $k \lt n$.
- Dan’s proof from Freudenthal suspension theorem (for suspension definition).

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

- 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.

- Mike’s proofs are in the book and Favonia formalized it.

- 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).