# nLab 3-sphere

Contents

### Context

#### Spheres

n-sphere

low dimensional n-spheres

# Contents

## 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.

## 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)

Last revised on July 27, 2020 at 12:33:12. See the history of this page for a list of all contributions to it.