The homotopy groups of spheres are a fundemental concept in algebraic topology. They tell you about homotopy classes of maps from spheres to other spheres which can be rephrased as the collection of different ways to attach a sphere to another sphere. The homotopy type of a CW complex is completely determined by the homotopy types of the attaching maps.
Here’s a quick reference for the state of the art on homotopy groups of spheres in HoTT. Everything listed here is also discussed on the page on Formalized Homotopy Theory.
In progress
Guillaume has proved that there exists an such that is . Given a computational interpretation, we could run this proof and check that is 2. Added June 2016: Brunerie now has a proof that , using cohomology calculations and a Gysin sequence argument.
At least one proof has been formalized
Peter L. has constructed 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.