nLab
3-sphere

Contents

Contents

Idea

The nn-sphere for n=3n = 3.

Properties

Isomorphisms

The underlying manifold of the special unitary group SU(2) happens to be isomorphic to the 3-sphere, hence also that of Spin(3).

The quotient of that by the binary icosahedral group is the Poincaré homology sphere.

Homotopy groups

The first few homotopy groups of the 3-sphere:

n=n =00112233445566778899101011111212
π n(S 3)=\pi_n(S^3) =*\ast0000\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2 12\mathbb{Z}_{12} 2\mathbb{Z}_{2} 2\mathbb{Z}_2 3\mathbb{Z}_3 15\mathbb{Z}_{15} 2\mathbb{Z}_2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2

e.g. Calabrese 16, for more see at homotopy groups of spheres.

References

Discussion of homotopy groups of spheres for the 3-sphere:

  • John Calabrese, The fourth homotopy group of the sphere, 2016 (pdf)

Discussion of 3-manifolds as branched covers of the 3-sphere:

  • J. Montesinos, A representation of closed orientable 3-manifolds as 3-fold branched coverings of S 3S^3, Bull. Amer. Math. Soc. 80 (1974), 845-846 (Euclid:1183535815)

Last revised on February 11, 2019 at 02:06:26. See the history of this page for a list of all contributions to it.