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 |
    https://i1.wp.com/upload.wikimedia.org/wikipedia/commons/3/3d/Wedding_rings.jpg
    crossed with a circle
  • a Möbius band is locally an interval [0,1] or vertical stick | but twisted once
    File:Möbius strip.jpg
  • a Hopf ring is locally an interval [0,1] or vertical stick | but twisted twice
    https://i2.wp.com/farm4.static.flickr.com/3016/2828881970_6e6d9c63ea.jpg
    https://i0.wp.com/images1.sw-cdn.net/model/picture/674x501_96765_97639_1338413385.jpg

    https://i1.wp.com/static2.ponoko.com/design_images/images/18512/25af8802-edd2-eea3-669e-461f706ee14f/hopf-ring_-_math_art.2500-390x390xffffff_product_page.jpg

    https://i0.wp.com/images1.sw-cdn.net/model/picture/674x501_96765_97640_1338413385.jpg

    https://i2.wp.com/cdnimg.visualizeus.com/thumbs/30/da/computer,generated,image,hopf,link,seifert,surface-30da215023e5f35242e8601a888dea06_h.jpg

  • Fibration 𝔽 → total space 𝔼 → 𝔹 base space
  • Hopf map: 𝕊¹→𝕊³→𝕊²
  • 𝕊⁰→𝕊¹→𝕊¹
  • 𝕊³→𝕊⁷→𝕊⁴
  • 𝕊→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.
    https://i0.wp.com/upload.wikimedia.org/wikipedia/commons/8/85/Stereographic_projection_in_3D.png
  • 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…”
    Pictures of the 3-sphere, or should I say the 4-ball? It’s a 4-dimensional circle.  Even though these drawings of it look completely sweet, I have a hard time parsing them logically. They’re stereographic projections of the hypersphere. All they’re trying to show is the shell of {4-D points that sum to 1}. That’s lists of length 4, containing numbers, whose items add up to 100%. Some members of the shell are  ∙ 10%  30%  30%  30%∙ 60%  20%  15%  5%∙ 0%  80%  0%  20%∙ 13%  47%  17%  23%∙ 47%  17%  23%  13%∙ 17%  23%  13%  47%∙ 0%  100%  0%  0%∙ 5%  5%  5%  85%   The hypersphere is just made up of 4-lists like that.    The 3-sphere was the object of the Poincaré Conjecture (which is no longer a conjecture). Deformations of this shell — this set of lists — are the only simply-connected 3-manifolds. Any other 3-manifold which doesn’t look holey or disjoint must be just some version of the hypersphere.
  • Some stuff I couldn’t see which was pretty important.
  • Minute 46. Rock out to the Hopf links.

(por Eddie Beck)

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