Posts tagged with fibres

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)

## Space

The word ‘space’ has acquired several meanings, which is what you would expect of such a sexy, primitive, metaphorically rich, eminently repurposeable concept.

1. Outer space, of course, is where cosmonauts, Hubble telescopes, television satellites, and aliens reside. It’s ℝ³, or something like that.

2. Grammatical spaces keep words apart. The `space` bar got a little more exercise than the `backspace` key while I was writing this list.

3. Non-printable area (space) is also free from ink or electronic text in newspapers: ad space. Would you like to buy one?

4. Space on my hard drive to store an exact digital replica of all my vinyl? This kind of space also applies to human memory capacity, computer RAM, and other electronic pulsings which seem rather more time-based than spatial & static.

5. Businessmen refer to competitive neighbourhoods: the online payments space; the self-help books category; the \$99-and-under motel space; and so on.

6. Space as distinct from time. Although cosmologists will tell you that spacetime is a pseudo-Riemannian manifold which looks locally like ℝ⁴, a geographer or ecologist will tell you that locally space looks like ℝ² (since we live solely on the surface of the Earth).

I believe the ℝ² view is also taken by programmers who geotag things (flickr photos, twitter tweets, 4square updates): second basement = 85th floor and canopy = rainforest floor as far as that’s concerned.

Both perspectives are valid. They’re just different ways of modelling “the world” with tuples. Is it surprising that cold, rigid, soulless mathematics allows for different, contradictory viewpoints? Time is like space in the grand scheme of things, but for life on Earth time-averages and space-averages are very different.

7. Parameter space. The first graphs one learns in school plot input x versus output ƒ(x).

But another kind of plot — like a solid liquid gas diagram

— plots input a versus input b, with the area coloured or labelled by output ƒ(x). (In the case of matter’s phases, the codomain of ƒ is the set {solid, liquid, gas, plasma} rather than the familiar .)

• When I push this lever, what happens? What about when I push that one?
• There are connections to Fourier spectrum.
8. Phase space. Paths, orbits, and trajectories taken through other spaces. Like the string of (x₍ᵤ₎,y₍ᵤ₎,z₍ᵤ₎)-coordinates that a water rocket takes across the lawn. Or the path of temperature (temp₍ᵤ₎) during a year in Bloomington.

Roger Penrose uses the example of the configuration space of a belt to explain that phases can happen on non-trivial manifolds. (A belt can take on as many configurations as a string, plus it can be twisted into a Moebius band, but if it’s twisted twice that’s the same as twisted zero times.)

[Sorry, I don’t have a Unicode character for subscript t, so I used u to represent the time-indexing of path variables. Maybe that’s better anyway, because time isn’t the only possible index.]

1. Personal space. I forgot personal space. Excuse me; pardon me.
2. Abstract spaces. These are best understood as ordered tuples, i.e. “Things plus the relationships and desired interpretation of those things.” The space—more like “the entire logical universe I’m going to be talking about here”—is supposed to contain EVERYTHING you need, in order to work with any of the parts. So for example to use a division sign ÷, the space must include numbers like and . (Or you could just do without the ÷ sign. You can make a ring that’s not a division ring; look it up.)

• A Banach space is made up of vectors (things that can be added together), is complete (there are enough things that infinite limit sequences make sense), with a notion of distance (norm), but not necessarily angle. Also two things can be 0 distance away from each other without being the same thing. (That’s unlike points in Euclidean space: (2,5,2) is the only thing 0 away from (2,5,2)).
• A group is complete in the sense that everything you need to do the operation is included. (But not complete in the way that Banach space is complete with respect to sequences converging. Geez, this terminology is overloaded with meanings!)

• A vector space is complete in the same way that a group is. In the abstract sense. Again, a vector is “anything that can be added together”. The vectors’ space completely brings together all the possible sums of any combination of summands.

For example, in a 2-space, if you had (1,0) and (0,1) in the space, you would need (1,1) so that the vector space could be complete. (You would also need other stuff.)

And if the vector space had a and b, it would need to contain a+b — whatever that is taken to mean — as well as a+b+b+(a+b)+a and so on. In jargon, “closed under addition”.
• A topological space (confusingly, sometimes called “a topology”) is made up of things, bundled together with the necessary overlap, intersection, union, superset, subset concepts so that “connectedness” makes sense.

• A Hilbert space has everything a Banach space does, plus the notion of “angle”. (Defining an inner product is as good as defining an angle, because you can infer angle from inner multiplication.) ℂ⁷ is a hilbert space, but the pair ({0, 1, 2}, + mod 2) is not.

• Euclidean space is a flat, rigid, stick-straight, all-joins-square Hilbert space.
• To recap that: vector space  Banach space  Hilbert space, where the  symbol means “is less structured than”.

Topological spaces can be even more unstructured than a vector space. Wikipedia explains all of the T0 T1 ⊰ T2 ⊰ T2.5  T3  T3.5  T4 ⊰ T5 ⊰ T6 progression which was thoroughly explored during the 20th century. (Those spaces differ in how separated “neighbours” are taken to be.)

I don’t mean to imply that these spaces can only be thought of as tuples: ({things}, operations). There are categorical ways to understand them which may be better. But don’t look at me; ask the ncatlab!

1. Lastly, sometimes ‘space’ just means a collection of related things, without necessarily specifying, like above, the tools and viewpoints that we take to their relationships.
• The space of all possible faces.
• The space of all possible boyfriends.
• The space of all possible songs.
• The space of all possible sentences.
• Qualia space, if you’re a theorist of consciousness.
• The space of all possible romantic relationships.
• The space of all possible computer programs of length 17239 bytes.
• Whatever space politics occupies. (And we could debate about that.)
• (consumption, leisure, utility) space
• The space of all possible strategy pairs.
• The space of all possible wealth distributions that sum to W.
• The space of all bounded functions.
• The space of all 8×8 matrices over the field ℤ₁₁.
• The space of all polynomials.
• The space of all continuous functions from [0,1] → [0,1].
• The space of all square integrable functions.
• The space of all bounded linear operators.
• The space of all possible models of ______.
• The space of all legal configurations of the Rubik’s cube.

(Some of these may be assumed to come packaged with a particular set of interpretations as in the previous ol:li.)