quaternionic projective line$\,\mathbb{H}P^1$
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:
Discussion of 3-manifolds as branched covers of the 3-sphere:
Classification of Riemannian orbifolds whose coarse underlying topological space is a 3-sphere:
Last revised on July 27, 2020 at 12:33:12. See the history of this page for a list of all contributions to it.