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`
• `𝕊``⁷→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)