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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.
In progress
Guillaume proved that there is some such that is . With a computational interpretation, we could run this proof and check that .
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 , together with , imply this by a long-exact-sequence argument, but this hasn’t been formalized.
Cohomology
Prove that is a spectrum (Eric?).
To do cohomology with finite coefficients, all we need is the boring work of defining 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 -connected.