Showing changes from revision #2 to #3:
Added | Removed | Changed
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.
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. such that is . With a computational interpretation, we could run this proof and check that .
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., together with , imply this by a long-exact-sequence argument, but this hasn’t been formalized.
Cohomology
Prove that K(G,n) is a spectrum (Eric?). 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. as an explicit group.
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.-connected.