Homotopy Type Theory
homotopy groups of spheres (Rev #15)


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.


n/k 0 1 2 3 4
0 π0(S0) π0(S1) π0(S2) π0(S3) π0(S4)
1 π1(S0) π1(S1) π1(S2) π1(S3) π1(S4)
2 π2(S0) π2(S1) π2(S2) π2(S3) π2(S4)
3 π3(S0) π3(S1) π3(S2) π3(S3) π3(S4)
4 π4(S0) π4(S1) π4(S2) π4(S3) π4(S4)

In progress

π 4(S 3)\pi_4(S^3)

  • Guillaume Brunerie has proved that there exists an nn such that π 4(S 3)\pi_4(S^3) is n\mathbb{Z}_n. Given a computational interpretation, we could run this proof and check that nn is 2. Added June 2016: Brunerie now has a proof that n=2n=2, using cohomology calculations and a Gysin sequence argument.

At least one proof has been formalized

π 3(S 2)\pi_3(S^2)

  • Peter LeFanu Lumsdaine 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 Brunerie’s proof that the total space of the Hopf fibration is S 3S^3, together with π n(S n)\pi_n(S^n), imply this by a long-exact-sequence argument.
  • This was formalized in Lean in 2016.

π n(S 2)=π n(S 3)\pi_n(S^2)=\pi_n(S^3) for n3n\geq 3

  • This follows from the Hopf fibration and long exact sequence of homotopy groups.
  • It was formalized in Lean in 2016.

Freudenthal Suspension Theorem

Implies π k(S n)=π k+1(S n+1)\pi_k(S^n) = \pi_{k+1}(S^{n+1}) whenever k2n2k \le 2n - 2

π n(S n)\pi_n(S^n)

π k(S n)\pi_k(S^n) for k<nk \lt n

π 2(S 2)\pi_2(S^2)

π 1(S 1)\pi_1(S^1)

category: homotopy theory

Revision on October 18, 2018 at 05:45:04 by Alexis Hazell?. See the history of this page for a list of all contributions to it.