Changes in the internal structure during a Topological transform

[arunrajagopal]

Changes in the internal structure during a "Topological transform"

Is there any field of topology which deals with the changes in internal structure of an object when it undergoes topological transform? If I'm transforming a cube into a sphere, is there any 'field of topology' which analyze the structural changes inside the cube. For example, in what way does the points defining the cube get affected. Can you please recommend me some books regarding this. Thanks in advance.

 

[Stephen Tashi]

If I'm transforming a cube into a sphere, is there any 'field of topology' which analyze the structural changes inside the cube. For example, in what way does the points defining the cube get affected.

That depends upon what you mean by "structure". I'll guess that you are interested in the type of structure that isn't really the subject of topology. It sounds like you are interested in very concrete matters like particular formulas for where a point (x,y,z) in a cube goes when the cube is transformed in a specific way into a sphere. Is that what you are asking about?

 

[homeomorphic]

Your question is very vague, so I have more than one answer.

By topological transformation, I assume you mean homeomorphism.

A cube is the same thing as a sphere in topology, proper. Differences between them aren't really studied in topology, except maybe differential topology. The only thing that is different about them from a differential topology standpoint is that a cube is not smoothly embedded because it has edges, but the sphere is smoothly embedded.

But if you're not doing differential topology or piecewise linear topology or something like that, if two things are homeomorphic, they are considered equivalent, so topology can't tell us anything about what you are talking about because topology can't see the difference between a sphere and cube, for example.

That was my first answer. Here is my second answer. There ARE branches of topology that study homeomorphisms themselves, which might be along the lines you are looking for. The homeomorphisms from a space to itself form a group under composition--actually, it's a topological group when endowed with the compact-open topology, for example. So, you can study that group, and in many cases, you want to consider a quotient group of that group. For example, you might study what's called "the mapping class group" of a surface. For that, you could try Farb and Margalit's book, A Primer on Mapping Class Groups, once you have the prerequisites down.

Another field that might get into this sort of thing a bit is dynamical systems. Again, you look at maps of a space to itself, and this case, you look at what happens when you keeping iterating the map over and over again.

 

[arunrajagopal]

That depends upon what you mean by "structure". I'll guess that you are interested in the type of structure that isn't really the subject of topology. It sounds like you are interested in very concrete matters like particular formulas for where a point (x,y,z) in a cube goes when the cube is transformed in a specific way into a sphere. Is that what you are asking about?

Your question is very vague, so I have more than one answer.

By topological transformation, I assume you mean homeomorphism.

A cube is the same thing as a sphere in topology, proper. Differences between them aren't really studied in topology, except maybe differential topology. The only thing that is different about them from a differential topology standpoint is that a cube is not smoothly embedded because it has edges, but the sphere is smoothly embedded.

But if you're not doing differential topology or piecewise linear topology or something like that, if two things are homeomorphic, they are considered equivalent, so topology can't tell us anything about what you are talking about because topology can't see the difference between a sphere and cube, for example.

That was my first answer. Here is my second answer. There ARE branches of topology that study homeomorphisms themselves, which might be along the lines you are looking for. The homeomorphisms from a space to itself form a group under composition--actually, it's a topological group when endowed with the compact-open topology, for example. So, you can study that group, and in many cases, you want to consider a quotient group of that group. For example, you might study what's called "the mapping class group" of a surface. For that, you could try Farb and Margalit's book, A Primer on Mapping Class Groups, once you have the prerequisites down.

Another field that might get into this sort of thing a bit is dynamical systems. Again, you look at maps of a space to itself, and this case, you look at what happens when you keeping iterating the map over and over again.

Thank you both for answering my query and understanding what I meant. Some days back I came across one theory which stated that position of at least one point would not be changed during a homeomorphic topological transform (correct me if I have understood this theory wrongly), if what I interpreted was correct, then I want some theory more broader than this (a homeomorphic topological property which deals with the internal structure). Sorry for being naive, I'm just a beginner in this subject.

 

[homeomorphic]

Thank you both for answering my query and understanding what I meant. Some days back I came across one theory which stated that position of at least one point would not be changed during a homeomorphic topological transform (correct me if I have understood this theory wrongly), if what I interpreted was correct, then I want some theory more broader than this (a homeomorphic topological property which deals with the internal structure). Sorry for being naive, I'm just a beginner in this subject.

You mean you found a theorem, not a theory that says that one point would be fixed. What you found is probably the Brouwer fixed point theorem. It says that if you map a disk continuously into itself (not necessary a homeomorphism, just a continuous map), then there is at least one point that stays fixed. So, if you stir a cup of coffee gently (without splashing or pushing the surface coffee go downward), at least one point is always in the spot where it started. It could be a different point for each moment in time.

http://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem

Fixed point theorems also occur outside topology. For example, the Banach fixed point theorem is not strictly topological, although you might cover it in a beginning point set topology class because it applies to metric spaces.

I guess you could just call this part of "fixed point theory". There is at least one journal bearing the name "fixed point theory". Presumably, it includes some topology, but isn't restricted to topology.

http://www.fixedpointtheoryandapplications.com/

I don't know anything about this subject, but one of the profs at my undergrad was a topologist who specialized in something called Nielson fixed point theory:

http://en.wikipedia.org/wiki/Nielsen_theory

 

[arunrajagopal]

You mean you found a theorem, not a theory that says that one point would be fixed. What you found is probably the Brouwer fixed point theorem. It says that if you map a disk continuously into itself (not necessary a homeomorphism, just a continuous map), then there is at least one point that stays fixed. So, if you stir a cup of coffee gently (without splashing or pushing the surface coffee go downward), at least one point is always in the spot where it started. It could be a different point for each moment in time.

http://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem

Fixed point theorems also occur outside topology. For example, the Banach fixed point theorem is not strictly topological, although you might cover it in a beginning point set topology class because it applies to metric spaces.

I guess you could just call this part of "fixed point theory". There is at least one journal bearing the name "fixed point theory". Presumably, it includes some topology, but isn't restricted to topology.

http://www.fixedpointtheoryandapplications.com/

I don't know anything about this subject, but one of the profs at my undergrad was a topologist who specialized in something called Nielson fixed point theory:

http://en.wikipedia.org/wiki/Nielsen_theory

Thank you for your response mate. This is the theorem I mentioned. Okay, let me try these links.