Topology

[geordief]

I have been thinking about this subject a bit (without any serious study) as I think it may be related to general relativity although I am not sure about that.

 

Anyway ,one of the examples that always comes up is the torus and I have noticed that you can make a torus by inserting a space in a 3d sphere or globe and then stretching the "remainder" of the sphere to that of a torus or indeed to any object that shares the same topology as the torus.

 

My question is this: Can we generalize from this method to create any exotic variety of topology we want by "making a space or gap" in the torus and subsequently in the new object so on ad infinitum?

 

If not ,is there any method or algorithm for creating new examples or types of topological spaces?

 

 

 

Just to repeat I know very little about this subject and I also have found it difficult to learn about it -so any corroboration , debunking or other explanation would be very welcome.

[swansont]

The topology description of this is the genus. "In layman's terms, it's the number of "holes" an object has"

https://en.wikipedia.org/wiki/Genus_(mathematics)

 

(and with this I have exhausted much of what I know about topology)

[Strange]

Definition of a topologist: someone who can't tell the difference between a coffee cup and a doughnut.

https://en.wikipedia.org/wiki/Homeomorphism

(I probably know even less about topology than swansont)

[geordief]

The topology description of this is the genus. "In layman's terms, it's the number of "holes" an object has"

https://en.wikipedia.org/wiki/Genus_(mathematics)

 

(and with this I have exhausted much of what I know about topology)

Clearly not required reading in Relativity then

 

"Number of holes" does seem to accord with my preconception ,though.

Definition of a topologist: someone who can't tell the difference between a coffee cup and a doughnut.

https://en.wikipedia.org/wiki/Homeomorphism

(I probably know even less about topology than swansont)

Very good There is a difference though. You cannot dunk a coffee cup.

[StringJunky]

Very good There is a difference though. You cannot dunk a coffee cup.

In physics there is someone working out the maths to do it because in a parallel world with the right physics dunking coffee cups into donuts will be entirely plausible

[geordief]

In physics there is someone working out the maths to do it because in a parallel world with the right physics dunking coffee cups into donuts will be entirely plausible

I don't approve of your spelling of "doughnut" but ,dammit (damn it?) it is accepted

[StringJunky]

I don't approve of your spelling of "doughnut" but ,dammit (damn it?) it is accepted

Oh no! I've been infected with an Americanism. I will write a hundred times 'doughnut' to purge myself.

[ajb]

May be you are describing surgery theory, which is 'cutting and pasting' smooth manifolds in order to build one with the topological properties you want.

 

Anyway, topology is important in general relativity, but the field equations themselves say just about nothing in this respect. The theory is about the local geometry.

 

In general relativity one usually specifies the topological manifold and then puts metric on it. There may be several 'global shapes' that are okay, for example you could look at compactifying Minkowski space-time. The local theory of special relativity will not be changed at all.

 

There are of course many 'topologies' that one could use. A topology is just a way of defining points that are 'close to each other'. The Alexandrov topology, which is related to the causal structure, is often used.

[StringJunky]

May be you are describing surgery theory, which is 'cutting and pasting' smooth manifolds in order to build one with the topological properties you want.

 

Anyway, topology is important in general relativity, but the field equations themselves say just about nothing in this respect. The theory is about the local geometry.

 

In general relativity one usually specifies the topological manifold and then puts metric on it. There may be several 'global shapes' that are okay, for example you could look at compactifying Minkowski space-time. The local theory of special relativity will not be changed at all.

 

There are of course many 'topologies' that one could use. A topology is just a way of defining points that are 'close to each other'. The Alexandrov topology, which is related to the causal structure, is often used.

It's not just about surfaces then?

[ajb]

It's not just about surfaces then?

Not at all, just that it is easy to 'see' using surfaces.

 

A topological space is a very general notion, there need not be any notion of dimension here other than that found in set theory. A topological manifold has more structure, locally looks like R^n.

 

For example, a discrete topological maifold has dimension 0 as it locally looks like R^0. This is clearly not a surface

[StringJunky]

Not at all, just that it is easy to 'see' using surfaces.

 

A topological space is a very general notion, there need not be any notion of dimension here other than that found in set theory. A topological manifold has more structure, locally looks like R^n.

 

For example, a discrete topological maifold has dimension 0 as it locally looks like R^0. This is clearly not a surface

Right. Thanks for the clarification.

[geordief]

May be you are describing surgery theory, which is 'cutting and pasting' smooth manifolds in order to build one with the topological properties you want.

 

 

I don't think so .If I have rightly understood the method you refer to is a bit like tree grafting.

 

The "method" I am suggesting is to start with a ,,one dimensional (I was thinking of a 3-dimensional .object but clearly if I can't start with 1 dimension then there is no point)

 

This one dimensional object (a straight line) is a set of points with perhaps 2 extremities .

 

Next we introduce an "exclusion zone " ( a gap) which forces the other points in the set to configure themselves around it in a continuous,organized way -a bit like the way the surface of running water deviates around a rock projecting from the water..They can stretch to any shape provided there is this gap .And the "gap" can increase to any size and shape also.

 

Once this new configuration is made it may be possible to introduce a second gap in the same way and that would give a new , more complex one dimensional topology.

 

This process would work in exactly the same way for any-dimensional object and the "gap" or "exclusion zone" would be correspondingly dimensional giving any amount of n-dimensional topologies.

 

Now I realize this may be seriously (and perhaps laughably ) flawed but that is what I had in mind

 

By the way I would be interested to know how we can define when 2 (or more?) points are close or closest to each other in these circumstances.

[blue89]

hi, now I continue on functional analysis. but of course it is related to topology. I do not think that this department is as difficult as it is being spoken. I am trying to modelize something to have modernized shape. @geordief your explanations includes very much key words to do or express something. but actually I do not want to give any information before my studies (project&articles) be published. because I do not think that there were so comprehensive or different studies available. and of course, (with high probability in meaning) we are not allowed to give anyone information without sceinetific corrects,or it must be so. I hope to explore something great.

 

have a succesfull life.

[wtf]

Anyway ,one of the examples that always comes up is the torus and I have noticed that you can make a torus by inserting a space in a 3d sphere or globe and then stretching the "remainder" of the sphere to that of a torus or indeed to any object that shares the same topology as the torus.

Your instincts are good. I think you'd benefit from picking up some mathematical concepts and terminology, and tightening up your exposition. I have some remarks along those lines.

 

* First, note that the sphere and the torus are both two-dimensional manifolds. That means that at each point, if you just look around a small area, it's "almost" a plane. In the same sense, a circle is a 1-dimensional manifold. You might take a look at that link. It's a key idea in the mathematics of studying shapes.

 

Also by the way, dimension is a very complicated topic, far more subtle than it appears. For example shapes can have fractional and even irrational dimension.

 

You are correct of course that a sphere is embedded in 3-space, and a circle is embedded in 2-space. That's the technical term. But we are careful to distinguish between a shape itself, like a sphere, and the space it's embedded in. There are in fact some 2-dimensional shapes that can't even be embedded in 3-space, for example the Klein bottle.

 

* Secondly, I I don't follow your construction. If you take a sphere and poke a hole in it, or remove just a single point, you can flatten out what's left to a circle or square in the plane; and in fact you can stretch it all the way out to be the entire plane. So in fact your idea to poke a hole in a sphere leaves you with the plane, topologically.

 

To make a torus out of a sphere you have to poke two holes, then glue the circumferences of the holes to each other. Is that what you are trying to say? It would be helpful to clarify this idea. Someone mentioned mathematical surgery and that's what this is.

 

Finally, without going into detail, the best way to describe a torus mathematically is as the Cartesian product of two circles. https://en.wikipedia.org/wiki/Torus

 

The "method" I am suggesting is to start with a ,,one dimensional (I was thinking of a 3-dimensional .object but clearly if I can't start with 1 dimension then there is no point)

 

This one dimensional object (a straight line) is a set of points with perhaps 2 extremities .

Your idea of extremities is unclear. Let me give you some examples.

 

In topology there's a notion of boundary points. For example we'd all agree that the open unit interval [0,1], the endpoints 0 and 1 are boundary points.

 

But what about the open unit interval (0,1)? By definition that set does not include its endpoints. But in topology we still consider 0 and 1 to be boundary points.

 

Now there's a problem for your idea, because the open unit interval (0,1) is topologically equivalent to the entire real line; which clearly does not have any boundary points. Having boundary points is not a topological property.

 

With this in mind, can you clarify what you mean by extremities?

 

 

Next we introduce an "exclusion zone " ( a gap) which forces the other points in the set to configure themselves around it in a continuous,organized way -a bit like the way the surface of running water deviates around a rock projecting from the water..They can stretch to any shape provided there is this gap .And the "gap" can increase to any size and shape also.

I confess I can't follow this at all. You just seem to be poking a hole in a surface or making a tear in it, but the concept of an exclusion zone is unclear. And in topology you can always stretch things as much as you want, so that "size and shape" are not relevant. A big square and a small circle are exactly the same topologically.

 

 

Once this new configuration is made it may be possible to introduce a second gap in the same way and that would give a new , more complex one dimensional topology.

No doubt you can put holes in things to change their topology. But this doesn't seem too significant. Rather it seems obvious. If I poke a hole in a sphere I no longer have a sphere.

 

I think your visualization is clear to you but not to me.

 

This process would work in exactly the same way for any-dimensional object and the "gap" or "exclusion zone" would be correspondingly dimensional giving any amount of n-dimensional topologies.

Well this is unclear to me in any dimension. Yes it's true that if you have a topological object and you put a hole in it you'll get some different topological object. But that's practically by definition, since topology is the study of properties that are invariant under stretching and moving but not tearing or putting holes in. So if you put a hole in a manifold you get a different manifold.

 

 

Now I realize this may be seriously (and perhaps laughably ) flawed but that is what I had in mind

I don't think it's flawed but it's vague. I think your instincts are good and you'd benefit from reading about manifolds and general topology; and also spending some time trying to tighten up your ideas.

[Strange]

* Secondly, I I don't follow your construction.

 

 

That is only because you are using the mathematical definition of sphere, while geordief is using the "lay" definition. So when he says one hole, he means two!

[geordief]

I don't think it's flawed but it's vague. I think your instincts are good and you'd benefit from reading about manifolds and general topology; and also spending some time trying to tighten up your ideas.

Thank you very much for your detailed attention to my post. I hesitate to answer before I am at least able understand the points you are making and how they apply to what I was trying to say.

 

I will have to give it a bit of time and then perhaps I will be able to answer.

 

I will only answer,though if I feel that I have gained a preliminary understanding of the points you have raised as it is not helpful to anyone for me to add confusion on top of my initial confusion.

 

It does look interesting though.

 

You did ask me a specific question or two but I still prefer to also leave that till later (if I feel that I can formulate a decent response)

[wtf]

Thank you very much for your detailed attention to my post. I hesitate to answer before I am at least able understand the points you are making and how they apply to what I was trying to say.

If anything I wrote is unclear I hope you'll give me the chance to make it clear.

 

I encourage you to ask all the questions you have. Please don't burden yourself with having to figure out basic topology before you feel worthy of asking a question.

 

I will have to give it a bit of time and then perhaps I will be able to answer.

 

I will only answer,though if I feel that I have gained a preliminary understanding of the points you have raised as it is not helpful to anyone for me to add confusion on top of my initial confusion.

Confusion is normal. Better to ask questions.

 

 

You did ask me a specific question or two but I still prefer to also leave that till later (if I feel that I can formulate a decent response)

You can ignore what I said about not understanding your torus construction. Strange pointed out that you are thinking of solid balls and I'm thinking of hollow spheres. By default when you say "sphere" in math it means a hollow one. So n-spheres always live in n+1 dimensional space. The 2-D sphere lives in 3-space. That's the official terminology for what it's worth. If it's solid, it's a ball. Also note that the standard torus is hollow as well. There's no special name for a solid torus.

 

Now that I understand that you are thinking of solid balls I agree with your construction. You are drilling a hole from one side of the earth to the other, and what's left is a torus. In fact you are left with a solid torus.

 

I must say I have no idea whether this idea generalizes to higher dimensions. Geometric visualization was never my strong point and your guess is as good as mine. I think of (hollow) torii as Cartesian products of circles. I find that easier to visualize than your idea of drilling through solid balls. But I'm sure the two approaches are equivalent.

Edited June 12, 2016 by wtf