Homotopy Type Theory
homotopy groups of spheres (Rev #14)


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.


$n\backslash k$ 0 1 2 3 4
0 [$\pi_0(S^0)$](#pinsn) [$\pi_0(S^1)$](#piksn) [$\pi_0(S^2)$](#piksn) [$\pi_0(S^3)$](#piksn) [$\pi_0(S^4)$](#piksn)
1 $\pi_1(S^0)$ [$\pi_1(S^1)$](#pinsn) [$\pi_1(S^2)$](#piksn) [$\pi_1(S^3)$](#piksn) [$\pi_1(S^4)$](#piksn)
2 $\pi_2(S^0)$ $\pi_2(S^1)$ [$\pi_2(S^2)$](#pinsn) [$\pi_2(S^3)$](#piksn) [$\pi_2(S^4)$](#piksn)
3 $\pi_3(S^0)$ $\pi_3(S^1)$ [$\pi_3(S^2)$](#pi3s2) [$\pi_3(S^3)$](#pinsn) [$\pi_3(S^4)$](#piksn)
4 $\pi_4(S^0)$ $\pi_4(S^1)$ [$\pi_4(S^2)$](#hopff) [$\pi_4(S^3)$](#pi4s3) [$\pi_4(S^4)$](#pinsn)

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 05:24:10 by Richard Williamson?. See the history of this page for a list of all contributions to it.