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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 onhomotopy 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 | 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) |
Implies whenever
Last revised on February 14, 2019 at 06:14:33. See the history of this page for a list of all contributions to it.