homotopy groups of spheres (Rev #12, changes)

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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~~Guillaume Brunerie~~$n$~~has proved that there exists an~~such that~~$n$~~$\pi_4(S^3)$~~such that~~is~~$\pi_4(S^3)$~~$\mathbb{Z}_n$~~is~~. Given a computational interpretation, we could run this proof and check that~~$\mathbb{Z}_n$~~$n$~~. Given a computational interpretation, we could run this proof and check that~~is 2. Added June 2016: Brunerie now has a proof that~~$n$~~$n=2$~~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~~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’s proof that the total space of the Hopf fibration is~~Guillaume Brunerie~~$S^3$~~’s proof that the total space of the Hopf fibration is~~, together with~~$S^3$~~$\pi_n(S^n)$~~, 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,~~Peter LeFanu Lumsdaine’s encode/decode-style proof, formalized by~~Dan,~~~~using~~~~a~~~~clever~~~~lemma~~~~about~~~~maps~~~~out~~~~of~~~~two~~~~n-connected~~~~types.~~Dan Licata, using a clever lemma about maps out of two n-connected types.

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

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

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

~~Mike’s proof by contractibility of total space of universal cover (~~Mike Shulman’s proof by contractibility of total space of universal cover (HoTT blog).~~Dan’s encode/decode-style proof (~~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’s proof using the flattening lemma.~~Guillaume Brunerie’s proof using the flattening lemma.

category: homotopy theory

Revision on October 17, 2018 at 10:47:52 by Ali Caglayan. See the history of this page for a list of all contributions to it.