How Do Transition Functions Define a Möbius Strip in Fiber Bundles?

In summary: Otherwise, I will do so.In summary, the Möbius strip is defined through "gluing instructions" between its two charts.
  • #1
cianfa72
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TL;DR Summary
Möbius strip fiber bundle defined through 'gluing instructions' between its two charts
Hi,

starting from this (old) thread
Fredrik said:
But suppose instead that the ##ϕ_{12}(b)## is the identity map on ##F## for all ##b## in ##X##, and the map ##f↦−f## for all ##b## in ##Y##. Then the bundle is a Möbius strip.
I'm a bit confused about the following: the transition function ##ϕ_{12}(b)## is defined just on the intersection ##U_1\cap U_2## and as said in that thread it actually amounts to the 'instructions' to glue together the two charts to obtain the Möbius strip.

Neverthless my doubt is: what about the 2 points in ##S^1## (the base space) not taken in account ? On that points (and for the corresponding fibers) we actually do not define any 'gluing instruction' ?
 
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  • #2
cianfa72 said:
Summary:: Möbius strip fiber bundle defined through 'gluing instructions' between its two charts
Any idea ? thanks
 
  • #3
The gluing instruction is to paste with a reflection, multiplication by ##-1## if you view the fibers as copies of the interval ##[-1,1]##.

If you use circles rather than intervals as the fibers then the gluing instruction to paste by reflection around an axis of the circle gives a fiber bundle whose fibers are circles. Geometrically it is a closed surface called the Klein Bottle.
 
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  • #4
lavinia said:
The gluing instruction is to paste with a reflection, multiplication by ##-1## if you view the fibers as copies of the interval ##[-1,1]##.
Sorry, as far as I understood, the intersection ##U_1\cap U_2## is made of two disjoint subsets (name it ##X## and ##Y##).

For Möbius strip the gluing instruction for fibers 'staying' onto base points belonging to ##A## are different from that 'staying' on point belonging on ##B##. Namely for fibers staying on ##A## points the gluing instruction is "paste without reflection" rather than with reflection (multiplication by -1) for fibers staying on ##B## points
 
  • #5
cianfa72 said:
Sorry, as far as I understood, the intersection ##U_1\cap U_2## is made of two disjoint subsets (name it ##X## and ##Y##).

For Möbius strip the gluing instruction for fibers 'staying' onto base points belonging to ##A## are different from that 'staying' on point belonging on ##B##. Namely for fibers staying on ##A## points the gluing instruction is "paste without reflection" rather than with reflection (multiplication by -1) for fibers staying on ##B## points
On one interval paste without reflection. On the other paste with reflection. Imagine making a Mobius band out of a strip of paper and it will be clear.
 
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  • #6
lavinia said:
On one interval paste without reflection. On the other paste with reflection. Imagine making a Mobius band and it will be clear.
That's clear to me. My concern is about the two points that do not belong to the intersection ##U_1\cap U_2## thus to neither ##X## nor ##Y##. For the fibers 'over' them (preimage under bundle projection of those two points in the base space) the gluing procedure does not apply at all ?
 
  • #7
cianfa72 said:
That's clear to me. My concern is about the two points that do not belong to the intersection ##U_1\cap U_2## thus to neither ##X## nor ##Y##. For the fibers 'over' them (preimage under bundle projection of those two points in the base space) the gluing procedure does not apply at all ?

I am not sure what two points you mean.

The picture I have is the circle covered by two over lapping segments that intersect in two disjoint open intervals.

On one interval the transition function is the identity and on the other it is reflection.
 
  • #8
lavinia said:
The picture I have is the circle covered by two over lapping segments that intersect in two disjoint open intervals.
These two disjoint open intervals do not cover the entire circle ##S^1## , right ?
So, what about fibers over the other (two) points belonging to ##S^1## ? That is my question !
 
  • #9
cianfa72 said:
These two disjoint open intervals do not cover the entire circle ##S^1## , right ?
So, what about fibers over the other (two) points belonging to ##S^1## ? That is my question !

The rest of the circle you do nothing. There is no transition function.
 
  • #10
lavinia said:
The rest of the circle you do nothing. There is no transition function.
As explained by @Fredrik in that thread, take the base ##B=S^1## and let ##U_1=S^1-\{(0,1)\}## and ##U_2=S^1-\{(0,-1)\}##.

Now ##U_1 \cap U_2## --made of the two disjoint subsets ##X## and ##Y## of ##S^1##-- does not cover enterely ##S^1## because ##\{(0,1)\}## and ##\{(0,-1)\}## are not included there.
 
  • #11
cianfa72 said:
As explained by @Fredrik in that thread, take the base ##B=S^1## and let ##U_1=S^1-\{(0,1)\}## and ##U_2=S^1-\{(0,-1)\}##.

Now ##U_1 \cap U_2## --made of the two disjoint subsets ##X## and ##Y## of ##S^1##-- does not cover enterely ##S^1## because ##\{(0,1)\}## and ##\{(0,-1)\}## are not included there.
Right.

Gluing happens only on intersections.
 
  • #12
I would suggest that, when quoting a problem to start a new thread, quote the whole problem.
 

Related to How Do Transition Functions Define a Möbius Strip in Fiber Bundles?

1. What is a Möbius strip?

A Möbius strip is a mathematical surface with only one side and one edge. It is created by taking a long strip of paper, giving it a half-twist, and then connecting the ends together. This results in a surface with only one continuous side and no distinguishable inside or outside.

2. How is a Möbius strip related to fiber bundles?

A Möbius strip can be thought of as a simple example of a fiber bundle, which is a mathematical concept used to describe the relationship between two spaces. In this case, the Möbius strip is the base space and the circle is the fiber. The strip is continuously twisted as it moves along the circle, creating a non-trivial fiber bundle.

3. What are the applications of Möbius strips as fiber bundles?

Möbius strips as fiber bundles have many applications in different fields of science, including physics, chemistry, and biology. They can be used to model and understand various phenomena, such as the behavior of light in optical fibers, the structure of DNA, and the properties of certain materials.

4. How is the Möbius strip as a fiber bundle different from a regular Möbius strip?

The main difference between a Möbius strip and a Möbius strip as a fiber bundle is the addition of the fiber component. This creates a more complex structure with a continuously varying twist, whereas a regular Möbius strip has a constant twist along its length.

5. Can the Möbius strip as a fiber bundle be visualized in 3D?

Yes, the Möbius strip as a fiber bundle can be visualized in 3D by using a three-dimensional model or animation. However, it is important to note that the Möbius strip itself is a two-dimensional surface, so any 3D visualization is an approximation of the true structure.

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