> 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•30m ago
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.
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.
skulk•1h ago
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•30m ago
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