What is a fibration?
by Niles Johnson
ℝ×ℝ = plane (infinitely big square)
ℝ×𝕊 = cylinder (infinitely long)
𝕊×𝕊 = torus
(Remember: 𝕊×𝕊≠𝕊² ! Because of the North Pole & South Pole. But neither does [0,1]×𝕊≠𝕊² since the 𝕊¹ needs to come together into points at either end 𝕊⁰.)
projection from 𝔸×𝔹 → 𝔹 is 𝔸×{b∈𝔹} ↦{b∈𝔹} for a given {b∈𝔹}. Here 𝔸 is the fibre.
a cylinder is locally an interval [0,1] or vertical stick | crossed with a circle
a Möbius band is locally an interval [0,1] or vertical stick | but twisted once
a Hopf ring is locally an interval [0,1] or vertical stick | but twisted twice
Fibration 𝔽 → total space 𝔼 → 𝔹 base space
Hopf map: 𝕊¹→𝕊³→𝕊²
𝕊⁰→𝕊¹→𝕊¹
𝕊³→𝕊⁷→𝕊⁴
𝕊⁷→𝕊¹⁵→𝕊⁸
That's it. That's all the fibrations of spheres by spheres over spheres.
Any quaternion q∈ℍ times its complement (flip signs on all the "weirdo" i,j,k terms) has magnitude one
q•k•q⁻¹ zeroes out the first term (reducing dimensionality from 4→3) and still the magnitude 1--meaning ℍ~ℝ⁴→ℝ³→𝕊².
boom.
Stereographic projection.
ℝ² is the same as the open unit disk (btw: disk is filled in whereas circle is not) with a point at ∞ -- think of "bubbling up"
"arctan is a great function to use for mapping the real line (without ±∞) down to a finite interval." (See also the video of why Bicontinuity is the right condition for topological sameness.)
"So, um, just imagine the three-sphere.... OK, that was easy. Now..."
Some stuff I couldn't see which was pretty important.
Minute 46. Rock out to the Hopf links.
(por Eddie Beck)