Homotopy Equivalences(bartoszmilewski.com)
bartoszmilewski.com
Homotopy Equivalences
https://bartoszmilewski.com/2025/06/20/weak-homotopy-equivalences/
25 comments
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
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
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.
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.
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
A Sphere is a Loop of Loops (Visualizing Homotopy Groups)
https://youtube.com/watch?v=CxGtAuJdjYI
"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
The homotopy hypothesis has something mystical about it.
https://math.ucr.edu/home/baez/homotopy/homotopy.pdf
in terry tao’s recent interview with lex fridman there’s an interesting bit on poincaré conjecture where he goes out of his way not to use these words.
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.
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.
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.
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.
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…
How he got famous is such a mystery…
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.
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.
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.
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.
I think we already have something quite similar available. There are AI podcasts that can be generated automatically (Google Notebooks, ElevenLabs, ...). If you input a transcript of an expert conversation and request it to be explained in simpler terms, I think you would closely match what you're looking for.
starting from knuth + pearl and evolving to potus and india pm is pretty amazing. he obviously brings something that people crave.
What a weird comment/expectation. He's an academic mathematician not an influencer. What exactly do you expect him to predict - whether the Riemann hypothesis will be proven?
i think the desire being expressed is to hear tao’s view on what is likely to happen on the various problems . as one of the most prominent mathematicians of our time his personal opinion would be valuable.
> what is likely to happen on the various problems
Pure math just doesn't work like that. Literally no one has any idea when XYZ problem will be proven - there's no way to know! It's not like building a stadium or a road or even a company , where you can see or anticipate the coming of a proof before it arrives. It could happen tomorrow or it could happen never. For example, of Hilbert's original 23 problems posed in 1900, three are still open.
Pure math just doesn't work like that. Literally no one has any idea when XYZ problem will be proven - there's no way to know! It's not like building a stadium or a road or even a company , where you can see or anticipate the coming of a proof before it arrives. It could happen tomorrow or it could happen never. For example, of Hilbert's original 23 problems posed in 1900, three are still open.
Some problems are characterized by many mathematicians as exceptionally challenging, to the point that there are currently no known methods or branches of mathematics capable of addressing them. For these issues, it is often proposed that a entirely new branch of mathematics needs to be developed to effectively address these types of challenges.
And then there are other problems for which progress has been made over the past decades. At times, a weaker version may have been proved. Occasionally, new ideas emerge in the field regarding how to advance effectively. An upper or lower bound may be identified and gradually improved over time.
Of course, while no one can predict the future, making an educated guess is of course interesting. I would appreciate it if Terry could share his thoughts on such estimates.
And then there are other problems for which progress has been made over the past decades. At times, a weaker version may have been proved. Occasionally, new ideas emerge in the field regarding how to advance effectively. An upper or lower bound may be identified and gradually improved over time.
Of course, while no one can predict the future, making an educated guess is of course interesting. I would appreciate it if Terry could share his thoughts on such estimates.
i agree. but if tao has any opinions on what will fall first or with what techniques, i would listen. tao himself says that some problems lack handholds to climb the sheer cliff . it would be nice to know what he views as handholds on the various problems that have them.
as an example here is knuth speculating or expressing opinions on P=NP
https://podcasts.apple.com/us/podcast/lex-fridman-podcast/id...
https://podcasts.apple.com/us/podcast/lex-fridman-podcast/id...
You would, but then others would also come out of the woodworks claiming how wrong Tao is (or was, and I told you so).
Even if that's true that's a lot of avoidable drama.
These kinds of conversation works great for pushing addictive engagement but I doubt it does a lot of good.
Even if that's true that's a lot of avoidable drama.
These kinds of conversation works great for pushing addictive engagement but I doubt it does a lot of good.
perhaps , but having some opinions shared is also a kind of guidance for people who don’t have access to the mathematics establishment. plus, if anyone can just shrug off that type of “told you so” nonsense it surely must be tao.
vaguely related : synthetic homotopies visualisation tool - https://github.com/marcinjangrzybowski/cubeViz2
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?