homotopy groups of spheres (Rev #11, changes)

Showing changes from revision #10 to #11:
Added | ~~Removed~~ | ~~Chan~~ged

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 | $\pi_0(S^0)$ | $\pi_0(S^1)$ | $\pi_0(S^2)$ | $\pi_0(S^3)$ | $\pi_0(S^4)$ |

1 | $\pi_1(S^0)$ | $\pi_1(S^1)$ | $\pi_1(S^2)$ | $\pi_1(S^3)$ | $\pi_1(S^4)$ |

2 | $\pi_2(S^0)$ | $\pi_2(S^1)$ | $\pi_2(S^2)$ | $\pi_2(S^3)$ | $\pi_2(S^4)$ |

3 | $\pi_3(S^0)$ | $\pi_3(S^1)$ | $\pi_3(S^2)$ | $\pi_3(S^3)$ | $\pi_3(S^4)$ |

4 | $\pi_4(S^0)$ | $\pi_4(S^1)$ | $\pi_4(S^2)$ | $\pi_4(S^3)$ | $\pi_4(S^4)$ |

$n\backslash k$ | 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) |

- Guillaume 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.

- Peter L. 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’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.

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

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

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

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

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

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

- Mike’s proof by contractibility of total space of universal cover (HoTT blog).
- Dan’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’s proof using the flattening lemma.

category: homotopy theory