3-sphere

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

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.

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*.

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)

Last revised on December 1, 2019 at 14:18:12. See the history of this page for a list of all contributions to it.