Homotopy Type Theory Formalized Homotopy Theory > history (Rev #3, changes)

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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)$\pi_4(S^3)$

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

π3(S2)$\pi_3(S^2)$

• 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 S3, together with π_n(Sn), imply this by a long-exact-sequence argument, but this hasn’t been formalized.$S^3$, together with $\pi_n(S^n)$, imply this by a long-exact-sequence argument, but this hasn’t been formalized.

Cohomology

• Prove that K(G,n) is a spectrum (Eric?).$K(G,n)$ 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.$\mathbb{Z}/p\mathbb{Z}$ as an explicit group.
• Calculate some cohomology groups.

At least one proof has been formalized

Whitehead’s theorem

• Dan’s proof for n-types.

K(G,n)

• Dan’s construction.

Freudenthal Suspension Theorem

Implies π_k(Sn) = π_k+1(Sn+1) whenever k <= 2n - 2$\pi_k(S^n) = \pi_{k+1}(S^{n+1})$ 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.$2n$-connected.

π_n(Sn)$\pi_n(S^n)$

• 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).

π_k(Sn) for k < n$\pi_k(Sn)$ for $k \lt n$

• Guillaume’s proof (see the book).
• Dan’s encode/decode-style proof for pi^1(S^2) only$\pi_1(S^2)$ only.
• Dan’s encode/decode-style proof for all k < n$k \lt n$.
• Dan’s proof from Freudenthal suspension theorem (for suspension definition).

π2(S2)$\pi_2(S^2)$

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

π1(S1)$\pi_1(S^1)$

• 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

• Chris/Peter/Jeremy’s development (link?).

Van Kampen

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

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).