Telescope Conjecture Disproved (Mathematics)

[W.D.Clinger]

homology

You can cut a sphere anywhere and get a circle, 1 map.

You can cut a doughnut in most ways to get a circle, but you can also cut it in another way to get something that's not a circle, more than one map.

This is not a matter of cutting a sphere or toroid. It's a matter of whether there are fundamentally different ways of placing a circle on a sphere or toroid.

• If you draw two different circles on a sphere, each of the circles can be deformed into the other.
• If you draw two different circles on a toroid, whether each of the circles can be deformed into the other depends on where you draw the circles. As illustrated by Wikipedia, it is possible to draw three different circles on a torus, none of which can be deformed into either of the other two.

That difference is related to the homological difference between a circle and a torus:

• If you draw any circle on a sphere, it can be shrunk to a point.
• If you draw a circle on a toroid, whether it can be shrunk to a point depends on how you draw the circle. Some circles can be shrunk to a point, but a circle that goes all the way around the hole cannot be shrunk to a point, nor can a circle that goes around the handle of a teacup (which is, topologically speaking, a torus) be shrunk to a point. (Wikipedia has pictures that explain this better than words.)

I'm confused, because it says, as sphenisc quoted "And if the space you’re mapping from is lower-dimensional than the space you’re mapping to (as in our example of the one-dimensional circle mapped onto a two-dimensional sphere), there is always only one map."

but it also says, about mapping a circle onto a toroid, that there are (at least) 2 distinct mappings.

Yes, I understand that part. It's the "And if the space you’re mapping from is lower-dimensional than the space you’re mapping to (as in our example of the one-dimensional circle mapped onto a two-dimensional sphere), there is always only one map." part which seems to disagree with there being 2 maps.

The whole statement was "If two spheres have the same dimension, there are always infinitely many maps between them. And if the space you’re mapping from is lower-dimensional than the space you’re mapping to (as in our example of the one-dimensional circle mapped onto a two-dimensional sphere), there is always only one map.",
the bit you quoted was referring to the highlighted i think, not in a general case of all shapes.

p0lka explained that correctly. The article really should have said "And if the sphere you're mapping from is lower-dimensional than the sphere you're mapping to, there is always only one map (up to homotopy)." Note that a circle is a 1-sphere.

This implies that, unlike the surface of a sphere, the surface of a torus is not two dimensional. But I am not topologist enough to know if this is true.

As Gord_in_Toronto has explained, a torus is two-dimensional. It is homeomorphic to S¹ × S¹, where S¹ is the 1-sphere, aka circle.

 

[arthwollipot]

As Gord_in_Toronto has explained, a torus is two-dimensional. It is homeomorphic to S¹ × S¹, where S¹ is the 1-sphere, aka circle.

...aaaand I've lost it again.

 

[W.D.Clinger]

...aaaand I've lost it again.

You can label the points of a circle by real numbers running from 0 (inclusive) to 2π (exclusive), which is to say we can identify the points of a circle with real numbers in the range [0,2π). We say the circle is 1-dimensional because each of its points corresponds to a single real number.

Because the circle is a 1-dimensional sphere, mathematicians write S¹ to mean a circle. As noted above, we can (for the purposes of this note) identify S¹ with the set of real numbers in the range [0,2π).

Now define T² as the Cartesian product T² = S¹ × S¹. Each point of T² corresponds to an ordered pair of real numbers (x,y). We say T² is 2-dimensional because each of its points corresponds to a pair of real numbers.

T² is a torus.

It is natural to ask whether it is possible to regard T² as a 1-dimensional space by defining a one-to-one correspondence between T² and some set of real numbers such as [0,2π). The answer to that question is that while it is possible to define such a one-to-one correspondence, all such one-to-one correspondences are discontinuous: nearby points of T² do not correspond to nearby points of [0,2π), and vice versa. To get a continuous one-to-one correspondence, we need to use pairs of real numbers. The dimension of a space is defined in terms of continuous one-to-one correspondences, which is why T² is not 1-dimensional.

Although what I wrote above should provide some intuition, it is not a mathematically rigorous definition of what it means for a topological space to be n-dimensional.

 

[arthwollipot]

Nope, you're talking well above my understanding again. That's okay, it's a lack of knowledge on my part, not any problem on yours. I have what I need to understand the basics of this result and why it is significant. Thanks though.

 

[Mike Helland]

Nope, you're talking well above my understanding again. That's okay, it's a lack of knowledge on my part, not any problem on yours. I have what I need to understand the basics of this result and why it is significant. Thanks though.

The torus is 3D, but it's surface is 2D.

Just like the surface of a 3D sphere is a 2D plane.

On the 2D plane that's a surface of a sphere, any two circles you draw can be morphed into each other.

On the 2D surface of the 3D torus, you could draw some circles that can't be morphed into each other, because there might be a hole in the way. One circle might be around a hole and the other circle might not be. So they can't be squished onto each other.

 

[Reformed Offlian]

However. I'm not a registered topologist, so what do I know.

My understanding of topology is only surface level.

I'll show myself out.

 

[Reformed Offlian]

...aaaand I've lost it again.

Me, too. Something something homoerotic something.

 

[arthwollipot]

Yes, you can draw a circle around the torus that "surrounds" the hole, like all the way around the outside rim, and that circle can't be mapped onto a circle that doesn't surround the hole, like one that goes from the outside rim to the inside surface of the hole and back. I can visualise that much (even without a diagram, though describing it in words is hard). The idea that the surface of a 3d object is 2d isn't a problem for me either, and I know why topologically speaking a coffee cup is the same as a doughnut.

Extending this idea to multiple dimensionalities is harder for me to visualise.

 

[Mike Helland]

Yes, you can draw a circle around the torus that "surrounds" the hole, like all the way around the outside rim, and that circle can't be mapped onto a circle that doesn't surround the hole, like one that goes from the outside rim to the inside surface of the hole and back. I can visualise that much (even without a diagram, though describing it in words is hard). The idea that the surface of a 3d object is 2d isn't a problem for me either, and I know why topologically speaking a coffee cup is the same as a doughnut.

Extending this idea to multiple dimensionalities is harder for me to visualise.

I think that's true for everyone.

Mathematicians deal with higher dimensions in math. Not pictures.

 

[arthwollipot]

I think that's true for everyone.

Mathematicians deal with higher dimensions in math. Not pictures.

And that, I think, is one main reason why I've never been able to handle the higher levels of mathematics. I find it hard to understand if I can't visualise it.

 

[p0lka]

Nope, you're talking well above my understanding again. That's okay, it's a lack of knowledge on my part, not any problem on yours. I have what I need to understand the basics of this result and why it is significant. Thanks though.

It was thought that as the gaps between dimensions got larger, the difference wouldn't get larger than larger, but this result says it can.

 

[Gord_in_Toronto]

It was thought that as the gaps between dimensions got larger, the difference wouldn't get larger than larger, but this result says it can.

And as far as significance: "What Is The Good of a Newborn Baby?" -- Ben Franklin.

 

[W.D.Clinger]

For most people, including most mathematicians, I think the main takeaway from this is

The mathematics of higher dimensions is even weirder and more complicated than anyone knew.​

As an example of why someone might care about that, this subforum houses a thread discussing ChatGPT, which is a large language model that operates by navigating a space with billions of dimensions.

 

[p0lka]

And as far as significance: "What Is The Good of a Newborn Baby?" -- Ben Franklin.

You can't **** a sphere, but you can **** a torus. Theres a difference.

 

[lionking]

My belief is that higher level mathematics is almost beyond learning. That the best students who spend decades studying it, may never get to understand these concepts. I think it’s more like art than science.

 

[Gord_in_Toronto]

You can't **** a sphere, but you can **** a torus. Theres a difference.

You're just not trying hard enough.

 

[p0lka]

And that, I think, is one main reason why I've never been able to handle the higher levels of mathematics. I find it hard to understand if I can't visualise it.

On a lighthearted note, if you can ride a bike you're doing 12th dimensional physics

 

[arthwollipot]

On a lighthearted note, if you can ride a bike you're doing 12th dimensional physics

Not consciously you're not.

Yes, technically every time you catch a ball you're doing calculus. But your body and brain do it automagically. If you stopped to actually calculate the trajectory of the ball...

 

[p0lka]

If you can deliberately miss catching the ball, then when you catch it, it's a calculation.

 

[W.D.Clinger]

This doesn't have anything to do with the telescope conjecture or homotopy, but it's an entertaining and somewhat disturbing historical anecdote.

Back in the late 1960s or early 1970s, some folks at the MIT Artificial Intelligence Laboratory programmed its humongously over-engineered robotic arm to play table tennis.

What do I mean by "humongously over-engineered"? As of the late 1970s, a Polaroid photograph thumb-tacked on the wall near that (by then retired) many-jointed robotic arm showed it configured as a chair suspended from a steel beam in the ceiling, with a graduate student sitting on it. That arm was powerful enough to throw her through the wall.​

That robotic arm did not play a particularly aggressive style of table tennis, but it was pretty good at returning whatever you hit at it...

...unless you hit the ball so the camera saw it in a certain quadrant. In that quadrant, the robotic arm reliably missed the ball.

Investigating that bug, the programmers found it wasn't a bug in their own code, but was a bug in the PDP-10's Fortran library. A certain trigonometric routine was flat-out incorrect in that quadrant. That error had gone undiscovered until the PDP-10 computer series was approaching the very end of its useful life.

Imagine all of the scientific papers whose calculations are called into question by that bug.

During the 1980s and 1990s, standard libraries shipped by both Apple and Microsoft contained at least half a dozen similar errors. I hope those libraries are more reliable now, but I don't really know.