nLab 3-sphere

Idea

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

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:

$n =$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$
$\pi_n(S^3) =$	$\ast$	$0$	$0$	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z}_{12}$	$\mathbb{Z}_{2}$	$\mathbb{Z}_2$	$\mathbb{Z}_3$	$\mathbb{Z}_{15}$	$\mathbb{Z}_2$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$

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

Related concepts

• Hopf fibration

• fuzzy 3-sphere

• 2-sphere

• 6-sphere

• 7-sphere

• n-sphere

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 $S^3$, Bull. Amer. Math. Soc. 80 (1974), 845-846 (Euclid:1183535815)

Classification of Riemannian orbifolds whose coarse underlying topological space is a 3-sphere:

• William Dunbar, Geometric orbifolds, Rev. Mat. Univ. Complutense Madr. 1, No.1-3, 67-99 (1988)