homotopy groups of spheres (Rev #16, 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/k | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

0 | π_{0}(S^{0}) |
π_{0}(S^{1}) |
π_{0}(S^{2}) |
π_{0}(S^{3}) |
π_{0}(S^{4}) |

1 | π_{1}(S^{0}) |
π_{1}(S^{1}) |
π_{1}(S^{2}) |
π_{1}(S^{3}) |
π_{1}(S^{4}) |

2 | π_{2}(S^{0}) |
π_{2}(S^{1}) |
π_{2}(S^{2}) |
π_{2}(S^{3}) |
π_{2}(S^{4}) |

3 | π_{3}(S^{0}) |
π_{3}(S^{1}) |
π_{3}(S^{2}) |
π_{3}(S^{3}) |
π_{3}(S^{4}) |

4 | π_{4}(S^{0}) |
π_{4}(S^{1}) |
π_{4}(S^{2}) |
π_{4}(S^{3}) |
π_{4}(S^{4}) |

- 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.

- 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.

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

- 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).

- 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).

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

- 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.

category: homotopy theory