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I'm not very well-versed in topology, but I know the basic concept of topological equivalence can be approximated by "counting the holes" in objects, in the sense that a sphere is different from a torus, but a torus is the same as a coffee mug because they have the same number of holes. I know this roughly approximates being able to smoothly deform one object into another. Can this be applied to interior "holes" as well, such as a hollowed-out sphere? Does the hollow center count as a "hole"? Is there a way to smoothly deform a hollow sphere into a torus or a sphere? If not, is a hollow sphere in a different topological equivalence class from both a sphere and a torus?

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    $\begingroup$ A donut is topologically equivalent to a coffee mug. A torus is topologically equivalent to the surface of a coffee mug, but not to the full volume of the coffee mug. $\endgroup$
    – Stef
    Nov 29, 2023 at 9:22
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    $\begingroup$ A torus has a hole, which you can drive through from infinity towards another infinity. The hole of a hollow sphere is not accessible from anywhere. So, without having any notion of topology, I can tell you that I heavily doubt both being equivalent. $\endgroup$
    – Dominique
    Nov 29, 2023 at 11:35
  • $\begingroup$ @Dominique That's a pretty huge hint that they are not equivalent, but look at the two blue shapes on that wikipedia page, would you say they are topologically equivalent or not? ;-) $\endgroup$
    – Stef
    Nov 29, 2023 at 16:25
  • $\begingroup$ @Stef My recollection from back when I took topology is that those are topologically equivalent, so that knot theorists have to get a bit more creative about their notion of equivalence (for example, the complements are not topologically equivalent). $\endgroup$
    – Teepeemm
    Nov 29, 2023 at 16:46
  • $\begingroup$ I think people are interpreting your "hollowed out sphere" as meaning the two-dimensional surface of a sphere. But I think maybe you mean a solid object with a void inside it, like a lump of clay with a bubble in it. (And similarly I think you mean a solid torus rather than the surface of a torus.) If that's what you mean it would be a good idea to edit the question to make it clear. $\endgroup$
    – N. Virgo
    Nov 30, 2023 at 7:12

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There are situations in which counting "holes" tells you about topological equivalence.

A torus has two "one dimensional" holes. There are two essentially different ways to draw a closed curve on a torus that you can't shrink to a point. It has one "two dimensional" hole: the volume it surrounds.

A sphere (what you call a "hollow sphere") has no one dimensional holes. Every closed curve can shrink to a point. It too has one two dimensional hole.

A ball (what you call a "solid sphere") has no holes in any dimension.

So those are three different topological objects.

Just counting holes doesn't tell the whole story. A disk has no holes in any dimension but it's not topologically equivalent to a ball.

And there are strange holes it's hard to imagine. If you start with a circular disk and glue together the endpoints of each diagonal (in principle - you can't do this in ordinary space) the resulting figure is the projective plane. It has a one dimensional hole with the unintuitive property that if you go around it twice you haven't gone around it at all.

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  • $\begingroup$ Technical quibble: the torus has no "2D holes" in the sense of homotopy (i.e. $\pi_2=0$) though its complement in the ambient space has two components (but if we go this route, that is also true of the sphere). $\endgroup$
    – coiso
    Nov 30, 2023 at 15:00
  • $\begingroup$ @coiso Fair point, not just a quibble. I think I'll leave the answer as is, though. $\endgroup$ Nov 30, 2023 at 15:21
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I'm not very well-versed in topology, but I know the basic concept of topological equivalence can be approximated by "counting the holes" in objects, [...]

That's not quite true. Two objects are topologically different if there is any "topological" difference between them. Counting the holes is one way to attempt to topologically distinguish between two things, but there are other ways—infinitely many ways, in fact.

Are there any topological differences between a (hollow) torus and a (hollow) sphere?

(Side note: in mathematics, names of shapes, such as "circle," "sphere," and "torus," almost always refer to the "hollow" versions of these shapes, not the "filled-in" version, unless otherwise stated.)

The answer is yes, we can find topological ways to distinguish them. For example, you can draw a loop around a torus which does not divide the torus into two parts. The same feat is impossible for a sphere: any loop you can possibly draw on a sphere will divide the sphere into two parts.

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There is no way to deform a "hollow sphere" (normally just called a sphere) into a torus, or what you call a "sphere", by which I'm guessing you mean a solid sphere, or disk.

There are lots of possible ways to see/explain this, but your remark about them being in different "topological equivalence classes" is a good way to think about it.

Worth remarking thay there are lots of different possible ways to think of topological spaces as equivalent, though, not just "counting holes".

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Certain characteristics are preserved by continuous transformations: these can be called "topological properties." A topological structure on a space allows for continuity to be defined even if there is no notion of "distance" on the space. So, for two topological spaces to be equivalent, ALL of their topological properties must be the same between them. One of these, as you mentioned, is the number/types of holes in a space. Other examples of such properties and quantities would be connectedness, compactness, the Euler number, universal covering space, and the list goes on. This corresponds intuitively to two spaces are equivalent if you can get from one to the other without "tearing" one space (making holes, ripping off a piece and leaving it separated, etc.).

I will denote the (hollow) sphere as $\mathbb{S}^2$ and the torus as $\mathbb{T}^2$.

Your intuition is correct in that both $\mathbb{S}^2$ and $\mathbb{T}^2$ have an "interior hole" (although this is not the proper mathematical term). Mathematically, we would say that "the second homology group of both spaces is the same." Intuitively. this just means that both spaces have a "two-dimensional hole."

However, since holes of ALL dimensions are topological invariants, it would have to be the case that $\mathbb{S}^2$ and $\mathbb{T}^2$ have not only the same "two-dimensional holes" but also the same "one-dimensional holes." Looking at the shapes/spaces, the hole you see when looking at a torus is a one-dimensional hole, but if you look at the sphere you'll see that it does not have such a hole. Thus, the two spaces are not "topologically equivalent" and so you are unable to deform one into the other without tearing/ripping the space.

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Intuitively easy argument: any closed simple curve will cut $S^2$ in two pieces. $T^2$ minus some closed simple curves is connected. See https://mathinstitutes.org/highlights/visualizing-pml

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In some ways, a hollow sphere can be thought of as having -1 holes - in that, if you put one hole in it, puncturing the sphere, the resulting object has no holes and is topologically equivalent to a disc.

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