Homotopy Type Theory
homotopy groups of spheres (Rev #12)


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\kn\backslash k01234
0π 0(S 0)\pi_0(S^0)π 0(S 1)\pi_0(S^1)π 0(S 2)\pi_0(S^2)π 0(S 3)\pi_0(S^3)π 0(S 4)\pi_0(S^4)
1π 1(S 0)\pi_1(S^0)π 1(S 1)\pi_1(S^1)π 1(S 2)\pi_1(S^2)π 1(S 3)\pi_1(S^3)π 1(S 4)\pi_1(S^4)
2π 2(S 0)\pi_2(S^0)π 2(S 1)\pi_2(S^1)π 2(S 2)\pi_2(S^2)π 2(S 3)\pi_2(S^3)π 2(S 4)\pi_2(S^4)
3π 3(S 0)\pi_3(S^0)π 3(S 1)\pi_3(S^1)π 3(S 2)\pi_3(S^2)π 3(S 3)\pi_3(S^3)π 3(S 4)\pi_3(S^4)
4π 4(S 0)\pi_4(S^0)π 4(S 1)\pi_4(S^1)π 4(S 2)\pi_4(S^2)π 4(S 3)\pi_4(S^3)π 4(S 4)\pi_4(S^4)

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 17, 2018 at 10:47:52 by Ali Caglayan. See the history of this page for a list of all contributions to it.