Fiber Bundle Basics: An Overview of Penrose's Twisted Bundles

In summary: He identifies this structure with a (the?) cylinder.I'm having trouble with his next example, which he calls a twisted bundle, which he identifies with a (the?) Möbius strip. There are pictures of Möbius strips. I can see what twisted means when it refers to a strip of paper, but what does twisted mean when it refers to a fibre bundle?I think I follow what a (cross-)section means, at least when the bundle is trivial: a function s : B --> E such that the projection p \; \circ \; s(x) = x. (This according to Wikipedia's more formal intro; Penrose's cross-section is only the image
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If I understand this characteristic class stuff, then it also seems that the tangent bundle of the 2 sphere can not have a 1 dimensional subbundle. For if so the tangent bundle would decompose into a Whitney sum of two line bundles and each would have zero Euler class because the sphere is simply connected. The Whitney sum formula for the Euler class would then imply that the Euler class of the 2 sphere is also zero which can not be because its Euler characteristic is 2.

More generally from the same kind of reasoning, it would seem that the tangent bundle of an even dimensional sphere does not have any proper subbundle.

I think that Arkajad was thinking that the Whitney sum of the Mobius line bundle over the circle is trivial. In this bundle, parallel translation around the circle brings a vector back to its negative. So one can not get a section through parallel translation, I guess. However, if one allows the vector to rotate 180 degrees as one moves around the circle once, you get a section. What does this mean about the bundle?
 
<h2>1. What is a fiber bundle?</h2><p>A fiber bundle is a mathematical concept that describes a space made up of two different types of objects, known as the base space and the fiber. The base space is a larger space that contains the fiber, which is a smaller space that is attached to each point in the base space. This allows for the study of relationships between different spaces and their properties.</p><h2>2. Who is Penrose and what is his contribution to fiber bundles?</h2><p>Sir Roger Penrose is a British mathematician and physicist who is known for his work in general relativity and cosmology. He is credited with developing the concept of twisted bundles, which are a type of fiber bundle that have a "twist" in the way the fiber is attached to the base space. This twist allows for a more flexible and dynamic understanding of the relationships between spaces.</p><h2>3. What are some real-world applications of fiber bundles?</h2><p>Fiber bundles have numerous applications in mathematics, physics, and engineering. They are used to describe electromagnetic fields, quantum mechanics, and even the structure of DNA. In engineering, fiber bundles are used to model complex systems, such as fluid dynamics and traffic flow.</p><h2>4. How do fiber bundles relate to topology?</h2><p>Fiber bundles are closely related to topology, which is the study of the properties of geometric spaces that are preserved under continuous deformations. In fact, fiber bundles are often used in topology to study the properties of spaces that are not easily described using traditional methods. The base space of a fiber bundle is often a topological space, and the fiber itself can have its own topological structure.</p><h2>5. Are there any limitations to using fiber bundles?</h2><p>While fiber bundles are a powerful tool for studying relationships between spaces, they do have some limitations. One limitation is that they can only be used to study spaces that are locally trivial, meaning that the properties of the fiber are the same at each point in the base space. Additionally, fiber bundles can become quite complex and difficult to visualize in higher dimensions, making them more challenging to work with.</p>

1. What is a fiber bundle?

A fiber bundle is a mathematical concept that describes a space made up of two different types of objects, known as the base space and the fiber. The base space is a larger space that contains the fiber, which is a smaller space that is attached to each point in the base space. This allows for the study of relationships between different spaces and their properties.

2. Who is Penrose and what is his contribution to fiber bundles?

Sir Roger Penrose is a British mathematician and physicist who is known for his work in general relativity and cosmology. He is credited with developing the concept of twisted bundles, which are a type of fiber bundle that have a "twist" in the way the fiber is attached to the base space. This twist allows for a more flexible and dynamic understanding of the relationships between spaces.

3. What are some real-world applications of fiber bundles?

Fiber bundles have numerous applications in mathematics, physics, and engineering. They are used to describe electromagnetic fields, quantum mechanics, and even the structure of DNA. In engineering, fiber bundles are used to model complex systems, such as fluid dynamics and traffic flow.

4. How do fiber bundles relate to topology?

Fiber bundles are closely related to topology, which is the study of the properties of geometric spaces that are preserved under continuous deformations. In fact, fiber bundles are often used in topology to study the properties of spaces that are not easily described using traditional methods. The base space of a fiber bundle is often a topological space, and the fiber itself can have its own topological structure.

5. Are there any limitations to using fiber bundles?

While fiber bundles are a powerful tool for studying relationships between spaces, they do have some limitations. One limitation is that they can only be used to study spaces that are locally trivial, meaning that the properties of the fiber are the same at each point in the base space. Additionally, fiber bundles can become quite complex and difficult to visualize in higher dimensions, making them more challenging to work with.

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