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Quaternions enable one to formally prove that SO(3) is not simply connected. Here's how: The path in quaternion space e^{\pi i t} is not a loop, but its image i
by ogogmad 6y ago
Quaternions enable one to formally prove that SO(3) is not simply connected. Here's how: The path in quaternion space e^{\pi i t} is not a loop, but its image in SO(3) under the covering map is a loop. Any contraction of the image loop would imply a contraction of its pre-image, which is impossible.
- ajkjk 6y agoyes, but that relies on knowing that the quaternions are such a cover, which is entirely non-obvious. An explanation that starts and ends with SO(3) without roaming into quaternions is necessary for intuition.