An n-manifold would just mean any n-dimensional manifold. These are very particular n-dimensional manifolds, namely, spheres.
Now of course, this is topology, so our equivalences are broad; but the thing these are all equivalent (homeomorphic) to is a sphere. Sure, you can take a more complicated shape that's equivalent to a sphere, but that complexity is incidental; the broad equivalences of topology let us ignore them. (Although, alternatively, they also let us turn the sphere into, say, a cube, if that's easier to think about, which often it is.)
also a hypercube is not a cube--it's an n-cube. otherwise this is just lazy pop science rhetoric to get the kids excited about their field (and eventually suppress wages in mathematics with their newly-supplied labor, degree in hand). except not even science, so even less important
I understand that these objects are topologically equivalent to n-spheres, but that doesn't make them n-spheres, let alone spheres proper. In fact, you point out that cubes and spheres are topologically equivalent despite zero spheres being cubes and zero cubes being spheres.
Looking at all the continuous functions from all dimensions of spheres into a particular topological space ends up giving rich algebraic information about the space. This is a cornerstone of algebraic topology. Turns out calculating this stuff for even just spheres can be subtle and mysterious.
Uniform distance matters not at all for any of this, but it does matter that your family of "spheres" be topologically equivalent to the round spheres.
Another model is you take the iterated suspensions starting with a pair of points (the zero sphere).
Yet another is to take boundaries of simplicies, or even cubes.
Topologists are those who are perfectly happy to call a paper towel tube an annulus.
>The n-dimensional unit sphere — called the n-sphere
LOL nevermind—I was right the first time. Thank you for confirming.
> Your failure to banish my suspicions despite effort makes me that much more confident in my original conclusion.
Side note: I've never considered this phenomenon in my life, and suddenly other people digging in in the face of evidence makes sense. A dubious "thank you" to you.
And if you don't become more confident in your idea after a well-orchestrated yet entirely failed attempt to destroy it, you are not a rational person lol. it's called trial by fire and it's older than i am
the sensationalist nature of the writing has generated a lot of discussion so I guess it has does its job
A "metal" in astronomy is everything other than hydrogen or helium.
Also, n-spheres are commonly just called spheres for brevity. So when I say “the fundamental problem of homotopy theory is to compute the homotopy groups of spheres,” I am referring to all homotopy groups of all (n-)spheres simultaneously.
> I don’t see how any of this is limited to spheres.
In fact you’re right, homotopy theory is not just limited to spheres! However, if we could readily compute the homotopy groups of spheres, then we would be able to compute the homotopy groups of any “reasonable space.” Here I’m referring to CW complexes [1] which are a very broad class of spaces that, up to homotopy equivalence, probably includes any space you care to think of. It is for this reason that the problem of computing the homotopy groups of spheres is so fundamental to homotopy theory more broadly.