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

Idea

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.

Table

$n\backslash k$	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

$\pi_4(S^3)$

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

At least one proof has been formalized

$\pi_3(S^2)$

• 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 $S^3$, together with $\pi_n(S^n)$, imply this by a long-exact-sequence argument.
• This was formalized in Lean in 2016.

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

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

Freudenthal Suspension Theorem

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

• Peter LeFanu Lumsdaine’s encode/decode-style proof, formalized by Dan Licata, using a clever lemma about maps out of two n-connected types.

$\pi_n(S^n)$

• Dan Licataand Guillaume Brunerie’s encode/decode-style proof using iterated loop spaces (for single-loop presentation).
• Guillaume Brunerie’s proof (for suspension definition).
• Dan Licata’s proof from Freudenthal suspension theorem (for suspension definition).

$\pi_k(S^n)$ for $k \lt n$

• Guillaume Brunerie’s proof (see the book).
• Dan Licata’s encode/decode-style proof for pi_1(S^2) only.
• Dan Licata’s encode/decode-style proof for all k < n (for single-loop presentation).
• Dan Licata’s proof from Freudenthal suspension theorem (for suspension definition).

$\pi_2(S^2)$

• Guillaume Brunerie’s proof using the total space the Hopf fibration.
• Dan Licata’s encode/decode-style proof.

$\pi_1(S^1)$

• Mike Shulman’s proof by contractibility of total space of universal cover (HoTT blog).
• Dan Licata’s encode/decode-style proof (HoTT blog). A paper mostly about the encode/decode-style proof, but also describing the relationship between the two.
• Guillaume Brunerie’s proof using the flattening lemma.