UF IAS 2012 Archive Formalized Homotopy Theory

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.
• 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

• 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.

π_n(Sn)

• 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

• 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 < n.
• Dan’s proof from Freudenthal suspension theorem (for suspension definition)

π2(S2)

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

π1(S1)

• 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

• Favionia’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).