[HN Gopher] Homotopy Equivalences
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Homotopy Equivalences
Author : ibobev
Score : 64 points
Date : 2025-06-20 09:56 UTC (3 days ago)
(HTM) web link (bartoszmilewski.com)
(TXT) w3m dump (bartoszmilewski.com)
| skulk wrote:
| > In fact such a 2-sphere can be wrapped around the core an
| arbitrary number of times.
|
| This is really hard for me to visualize. What does it look like
| for a 2-sphere to wrap around the core multiple times? Also, I
| would have expected it to be able to wrap around in multiple ways
| since there are more dimensions here, leading to pi^2(b^3 \ {0})
| = Z^2. How would one even prove that this isn't the case?
| semolinapudding wrote:
| There is a nice illustration of a 2-sphere wrapped twice around
| another 2-sphere on the Wikipedia article for the homotopy
| groups of spheres [0].
|
| Now, there are many ways of proving that there is only one way
| (up to homotopy) of wrapping a 2-sphere n times around another
| 2-sphere, but all of them are fairly involved. The simplest
| proof comes from an analysis of the Hopf fibration, which
| roughly describes a relation between the 1-sphere, the 2-sphere
| and the 3-sphere [1]. Other than this, it follows from the
| theory of degrees for continuous mappings, or from the
| Freudenthal suspension theorem and some basic homological
| computations.
|
| [0]
| https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres#/me...
|
| [1] https://en.wikipedia.org/wiki/Hopf_fibration
| pfortuny wrote:
| This answer is probably a bit convoluted and possible
| erroneous. Assume the Earth has radius 2. Use coordinates (t,z)
| to denote "longitude" and "latitude from the North pole". Thus
| (0,0) is the North Pole, (0, pi/2) is the Greenwich equatorial
| point and (0, pi) is the South pole.
|
| You can have "two" spheres wrapped within the Earth with the
| following parametrization. Using a first coordinate r to denote
| the distance to the Earth's center, so that (1,t,z) denotes the
| points in the sphere of radius 1:
|
| (a,b)-> (1+cos(b)/2, a,b), for a,b in the interval [0,2pi].
|
| Those are not proper spheres (the radius changes) but the
| surface so parametrized is homotopic to a sphere "counted two
| times".
|
| It is not possible to have a warped sphere which does not cross
| itself, as far as I can tell (but I might be wrong).
|
| The wikipedia image linked by a sibling comment did not help
| me...
|
| ETA: the issue is not the dimension (2) of your spheres but the
| codimension (1) inside the object, and the fact that you have
| only removed the center of the main sphere. I think (caveat
| emptor) that if you remove 2 points form the solid sphere, you
| get Z^2. Similar to the case of surfaces and holes.
| lying4fun wrote:
| this great visualisation of homotopy groups might be helpful
|
| A Sphere is a Loop of Loops (Visualizing Homotopy Groups)
|
| https://youtube.com/watch?v=CxGtAuJdjYI
| coderatlarge wrote:
| in terry tao's recent interview with lex fridman there's an
| interesting bit on poincare conjecture where he goes out of his
| way not to use these words.
| randomtoast wrote:
| It's a good (and long) interview, and I genuinely enjoyed it.
| Terry Tao comes across as a truly nice person. However, I
| noticed that he tends to be somewhat non-committal in his
| responses. For each question posed, he provides thorough
| explanations that most with a basic understanding of math can
| follow. Nevertheless, he rarely makes predictions or offers his
| opinion. He frequently ends with a remark such as, "Yes, well,
| it's a challenging problem."
|
| I completely understand where he is coming from. While it's
| true that "we don't know what we don't know", I would
| appreciate hearing more about his (opinionated) thoughts
| regarding the topics discussed during the interview.
| williamstein wrote:
| Fascinating observation. Maybe he is better at research
| partly by being disciplined to not have such opinions. Having
| an opinion can bias one's approach to a problem, making it
| harder to solve.
| coderatlarge wrote:
| maybe a more mathematical interviewer could hove drawn out
| more predictions. i appreciate lex for having invited tao.
| i hope he manages to convince perelman.
| xanderlewis wrote:
| Just about anyone would be a more mathematical
| interviewer than Fridman. Even when it comes to CS, it's
| blatantly obvious he doesn't know what he's talking
| about.
|
| How he got famous is such a mystery...
| coderatlarge wrote:
| starting from knuth + pearl and evolving to potus and
| india pm is pretty amazing. he obviously brings something
| that people crave.
| randomtoast wrote:
| I think Lex is prepping his interviews very well. He will
| ask questions that address the areas in which the
| interviewee is an expert in. However, you begin to
| realize that he often struggles to ask follow-up
| questions that are relevant to what the interviewee has
| shared on various topics.
|
| This differs in other podcasts. For example, Sean
| Carroll, a theoretical physicist, conducts interviews
| with colleagues, who are also theoretical physicists.
| This enables him to engage in a meaningful conversation
| with the person being interviewed. When both parties
| strive to use language that a wider audience can
| understand, it truly becomes enjoyable.
| Syntonicles wrote:
| It's a lot to ask of two experts to also be excellent
| communicators for an audience that may struggle to follow
| along.
|
| I wonder if a potential application of LLMS could be:
| have two experts have a really interesting but dense
| conversation with each other, and then translate the
| conversation into simpler language with interjections for
| explanations.
|
| It may not be enjoyable for the most general audience,
| but it would scratch an itch for some of us.
| hackandthink wrote:
| "Grothendieck conjectured that the infinity groupoid captures all
| information about a topological space up to weak homotopy
| equivalence"
|
| The homotopy hypothesis has something mystical about it.
|
| https://math.ucr.edu/home/baez/homotopy/homotopy.pdf
| m_j_g wrote:
| vaguely related : synthetic homotopies visualisation tool -
| https://github.com/marcinjangrzybowski/cubeViz2
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