*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)

(Source: http://www.youtube.com/)

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