3维旋转群SO(3)不是单连通的

hbghlyj

gathering4gardner

3维旋转群SO(3)不是单连通的证明。从绕垂直轴旋转 180 度开始，然后不断转动旋转轴，直到指向下方。由于顺时针旋转 180 度与逆时针旋转 180 度相同，因此我们的路径是闭合的（起点、终点相同）。 如果 SO(3) 是单连通的，我们将能够连续地收缩这个闭环，直到我们得到一个点。 然而，无论我们如何调整这个旋转路径，我们的旋转轴必须在某个点是水平的。 因此，它不能连续收缩成一个点，所以SO(3)不是单连通的。

As it turns out, the space of rotations isn’t simply connected – unless we allow ourselves to represent each rotation twice, in which case we get the quaternions. To see this, consider the cycle given by starting with a 180-degree rotation around the vertical axis, and then continuously turning the axis until it points downward. Since a 180-degree clockwise rotation is the same as a 180-degree counterclockwise rotation, our path starts and ends at the same rotation. If the space of rotations were simply connected, we would be able to smoothly adjust and contract this loop until we get a single point. However, no matter how we adjust this path of rotations, our axis of rotation must at some point be horizontal. Therefore, we cannot turn this path into a single point, and so our space is not simply connected.

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hbghlyj

Dicyclic group是四元数群的由$x=e^{\frac {i\pi }{n}}=\cos {\frac {\pi }{n}}+i\sin {\frac {\pi }{n}}$和$y=j$生成的子群.
$x$对应于关于$(1,0,0)$旋转$2π/n$
$y=j$对应于关于$(0,1,0)$旋转$π$
当$n=3$时$o(x)=o(y)=o(xy)=4$ (但是$y$和$xy$对应的3维旋转的阶是2 因为SU(2)同构于SO(3)的double cover!)
$$y x=j e^{2 i \pi / 3}=j\left(-\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)=\left(-\frac{1}{2} j-\frac{\sqrt{3}}{2} k\right)=\left(-\frac{1}{2}-\frac{\sqrt{3}}{2} i\right) j=e^{4 \pi i / 3} j=x^{-1} y$$