nLab homotopy groups of spheres in HoTT

Idea

This entry is about homotopy groups of spheres formalized in homotopy type theory (HoTT).

The homotopy groups of spheres are a fundamental concept in algebraic topology and homotopy theory. They are the homotopy classes of maps between $n$-spheres, as $n$ varies. These may equivalently be understood as collection of different ways to attach spheres to each other. For instance, the homotopy type of a CW complex is completely determined by the homotopy types of the attaching maps.

The following is a quick reference for the state of the art on formalizing and computing homotopy groups of spheres in homotopy type theory.

Table

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)

Formalized proofs

The following is a list of results for which at least one proof has been formalized in HoTT.

Calculation of $\pi_4(S^3)$

• Brunerie 2016 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.

See also at first stable homotopy group of spheres.

Calculation of $\pi_3(S^2)$

• 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.

• Brunerie 2016, 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.

See also at second stable homotopy group of spheres.

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.

Calculation of $\pi_n(S^n)$

• Dan Licata and 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).

See also at Hopf degree theorem.

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

• Brunerie 2016 (see the HoTT 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).

Calculation of $\pi_2(S^2)$

• Brunerie 2016, using the total space the Hopf fibration.
• Dan Licata‘s encode/decode-style proof.

Calculation of $\pi_1(S^1)$

• 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.

See also at circle type for more.

Related entries

• first stable homotopy group of spheres

• second stable homotopy group of spheres

• third stable homotopy group of spheres

• synthetic homotopy theory

• Hopf construction in homotopy type theory

• cohomology in homotopy type theory

References

• Guillaume Brunerie, On the homotopy groups of spheres in homotopy type theory $[$arxiv:1606.05916$]$

• Axel Ljungström, The Brunerie Number Is -2 (June 2022) $[$blog entry$]$

category: homotopy theory