- #1

- 624

- 11

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- #1

- 624

- 11

- #2

- 18,265

- 20,227

So the short answer is: A fiber bundle is all of them (total space E, base space X, projection map π, fibers F). The crucial part in its understanding is, that it is in general not globally E = F x X but only locally. To call it union is even less exact.

- #3

Science Advisor

Gold Member

- 6,376

- 8,736

I think to complete the bundle the remaining instructions are the gluing maps up to isotopy ( maybe homotopy, not sure). You have a line " floating" about each point in the circle, by definition. At some point, in order to close the circle, two lines will have to come together " glued " to each other in certain ways. The "certain ways" is a choice of map between two lines , up to isotopy/homotopy. So, for each isotopy class of maps you have a choice of bundle up to bundle morphism. One class of self-maps is given by the identity, which gives you the cylinder: do the same parametrization in each line, say f(t)=t; 0<t<1 and map each t to itself, or the map t--> 1-t , which gives you the Mobius . I think finding the isotopy group of the Real line should do it here: either order-preserving or order-reversing are the two classes.

So the short answer is: A fiber bundle is all of them (total space E, base space X, projection map π, fibers F). The crucial part in its understanding is, that it is in general not globally E = F x X but only locally. To call it union is even less exact.

Last edited:

- #4

- 18,265

- 20,227

- #5

Science Advisor

Gold Member

- 6,376

- 8,736

Sorry, I don't know what you mean by this. I was addressing his example of fiber bundle fiber ##[-1,1]## over ##\mathbb S^1 ##.. Sorry for quoting you instead.As far as I see it, this is not necessary to have a fiber bundle. .

- #6

- 18,265

- 20,227

A fiber bundle is two topological spaces, a continuous projection with fibers as its preimages. I don't see where gluing should come into play rather than by special cases for applications. The entire topic is certainly inspired by tangent bundles and atlases. But for the pure definition, I don't think geometric concepts (lines, homotopies etc.) are needed.Sorry, I don't know what you mean by this.

Uh yes, sure.I was addressing his example of fiber bundle fiber ##[-1,1]## over ##\mathbb S^1 ##.. Sorry for quoting you instead.

- #7

Science Advisor

Gold Member

- 6,376

- 8,736

A fiber bundle is two topological spaces, a continuous projection with fibers as its preimages. I don't see where gluing should come into play rather than by special cases for applications. The entire topic is certainly inspired by tangent bundles and atlases. But for the pure definition, I don't think geometric concepts (lines, homotopies etc.) are needed.

.

I was referring to the specific case of the OP : the fibers circle around the base and must ultimately be glued together at some point.

- #8

Science Advisor

Gold Member

- 6,376

- 8,736

Sorry for my confusion, Fresh_Meister

Share:

- Replies
- 1

- Views
- 683

- Replies
- 3

- Views
- 2K

- Replies
- 1

- Views
- 1K

- Replies
- 1

- Views
- 2K

- Replies
- 12

- Views
- 2K

- Replies
- 5

- Views
- 6K

- Replies
- 18

- Views
- 2K

- Replies
- 0

- Views
- 331

- Replies
- 8

- Views
- 3K

- Replies
- 1

- Views
- 852