Homotopy Equivalences
7 comments
·June 20, 2025skulk
semolinapudding
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...
coderatlarge
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.
randomtoast
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
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
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.
> 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?