# nLab homotopy groups of spheres in HoTT

Contents

### Context

#### Spheres

n-sphere

low dimensional n-spheres

# Contents

## 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.

### Calculuation 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.

### Calculuation 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.

### Freudenthal Suspension Theorem

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