I recall learning several years ago that every three dimensional figure can be described as being topologically equivalent to a sphere with some number of holes through it. For example, a donut, a rubber band, and a teacup are both equivalent to a sphere with one hole through it. And others have two holes, and others three, and so on.
Well, that’s as much as I remember. I’m sure there’s more to it than that. And that’s why I’m posting this question.
I just had an occasion to imagine a sphere with, not just holes through it, but branching tunnels. For example, the one I was imagining was a sphere with three openings into it, where all three of the tunnels corresponding to these openings meet in the middle. So you get sort of a Y-shaped tunnel structure inside the sphere.
This doesn’t seem amenable to the process of simply counting up the number of holes. Something more complicated is going on–I think.
Am I right to think so? If so, how are such branching tunnels accomodated into the scheme of equivalency I referred to above?
I believe a sphere with a Y-shaped tunnel would be topologically equivalent to a torus with a hole drilled through the side. I don’t know if that helps. The process includes trying to imagine what would happen if the tunnels or holes were reduced to their shortest form.
Things are getting hard for me to visualize, but still, I don’t see how that article can be right.
I wish I could draw diagrams on this forum somehow.
Take a normal donut. Imagine it laying flat on a plane. The bottom surface of the donut is in contact with the plane at a circular path going around the bottom of the donut. Call that path P1.
Now imagine a circle around the donut “the other way,” i.e., starting at the outside, going around the top to the inside of the donut, then back around the bottom to the outside. This path forms a ring around the donut in a way different than did P1. Call this new path P2.
There is no way to smoothly move or stretch P1 such that it comes to coincide with P2. P1 and P2 therefore define mutually exclusive classes of paths around the donut. To my recollection there are just two such classes, and this is what it means to say the donut has a degree (or whatever they call it) of two.
Now take the donut with a “hole in a hole” as described above. There is an analogue to P1 on this figure, and an analogue to P2 as well. As far as I can tell, these are still mutually exclusive. And furthermore, it seems to me there’s a third path that can’t be transformed into either of the others. (If I’m wrong, this is probably where I’m wrong.) I’m thinking of the path that goes around the outermost edge of the donut–the circle on its outer surface which has the largest possible diameter assuming a “canonical donut,”–and which goes into the “hole in a hole,” then traces around the circle defining the main hole of the donut, then comes back out the “hole in a hole” and continues around the outermost edge back to the beginning. Call this P3.
P1 P2 and P3 appear to me to define three mutually exclusive classes of paths. But it’s really hard for me to visualize what’s going on with this shape, so there’s every chance I’m wrong here. I would like it if someone could explain how.
Okay, I just figured out how to transform my P3 into a path of type P2, if I’m allowed to make the path intersect itself at some points during the transformation. Am I so allowed?
As far as I can tell, your analysis in the previous post looks correct. And I think a sphere with a Y-shaped hole, and a sphere with two separate holes, can also each have 3 non-equivalent paths.
For the sphere with two separate holes, imagine a path (like a loop of thread) P1 tied through one of the holes, a path P2 tied through the other hole, and a third path P3 through both holes.
For the sphere with the Y-shaped hole, imagine paths through each possible pair of the three branches of the Y,
Depends if you’re talking about homotopy or homology. (Warning: jargon ahead. Also, take it with a grain of salt, since I don’t have my topology books here at home and my topology courses are some years behind me.)
The first homotopy group of a manifold (aka its fundamental group) is essentially the set of all directed paths on the manifold, with two paths defined as being equivalent if and only if they can be smoothly deformed into one another; the first homology group is the set of all loops in the space that differ be the boundary of some area. Letting a loop intersect itself & pinch itself off is fine if you care only about the loop structure; but if you care about the loop actually having a direction to it, then you can’t necessarily do so.
I seem to recall that the first homology group of a manifold is the same as the “Abelianization” of its first homotopy group, i.e., the quotient of the fundamental group by its commutator subgroup. This doesn’t make a difference for the one-holed torus, since its fundamental group (Z x Z) is already Abelian; but the fundamental group of the two-holed torus is non-Abelian, and so “Abelianizing” it yields a different group (Z x Z x Z x Z, IIRC.) This is why you can get certain paths to agree only if you let them intersect on the two-torus, whereas such shenanigans weren’t necessary on the one-torus.
BTW, the above results imply that there are actually four non-equivalent loops on the two-torus, even without taking directionality into account. (With directionality, there are an infinite number of loops.) See Figure (b) on this page for a picture of them on the two-torus.