Twisted Spaces: Fiber Bundles

Jan 28 2007 Published by under topology

It's been a while since I've written a topology post. Rest assured - there's plenty more topology to come. For instance, today, I'm going to talk about something called a fiber bundle. I like to say that a fiber bundle is a cross between a product and a manifold. (There's a bit of a geeky pun in there, but it's too pathetic to explain.)

The idea of a fiber bundle is very similar to the idea of a manifold. Remember, a manifold is a topological space where every point is inside of a neighborhood that appears to be euclidean, but the space as a whole may be very non-euclidean. There are all sorts of interesting things that you can do in a manifold because of that property of being locally almost-euclidean - things like calculus.

A fiber bundle is based on a similar sort of idea: a local property that does not necessarily hold globally - but instead the local property being a property of individual points, it's based on a property of regions of the space.

So what is a fiber bundle, and why should we care? It's something that looks almost like a product of two topological spaces. The space can be divided into regions, each of which is a small piece of a product space - but the space as a whole may be twisted in all sorts of ways that would be impossible for a true product space.

For example, what's the difference between a cylinder and a mobius strip? They're both formed by taking a square, and joining opposite edges. But the cylinder is a product space - it's the product of a line and a circle; the mobius strip is not a product space, because it's got a twist in it. But in most respects, it behaves very much like a product space - in particular, in any small local region, it's indistinguishable from the cylinder. The mobius strip is a fiber bundle.

To get into the formal definition, when we define a fiber bundle, we talk about three different topological spaces, and a mapping function. The three spaces are usually called B (the base space), E (the total space, and F (the Fiber); and the mapping function is called π (the projection map). π : E → B is a continuous onto function mapping from the total space to the base space. Every small region of the total space E looks like a region of the product of the base space B and the fiber F. (The onto part means that for every point b ∈ B, there is at least one point e ∈ E where π(e)=b.)

As usual, there's a bit more to it. The projection map and the spaces need to satisfy some conditions that define what it means to have the property of locally looking like a product space. To understand this, it will help to remember what a product space looks like. A product space is a space formed from a pair (or a collection) of component spaces, with continuous projection maps from the product space to each of its components. We're doing the same sort of thing here with the fiber bundle - except that we're only going to look at one projection function, and we're going to let it be a bit funky.

Here's the tricky part. Suppose we have a point p in B. For the projection function to work property, there must be an open neighborhood N including p where π-1 is homeomorphic to N × F and the projection function from N × F to N correspond. That's actually easier to understand using a diagram in the category of topological spaces. (It's things like this that make me appreciate category theory; as a primarily visual thinker, I find it much easier to grasp the meaning of this constraint by thinking about the diagram.) In the following diagram, h is a homeomorphism from π-1(N) to N×F, and pN is the projection function from N×F to N. The requirement on the fiber bundle's projection function is that the following diagram commutes:


bundle-cat.jpg

So now that we have both an intuitive and a formal sense of what a fiber bundle is, let's look at a couple of examples. I'm going to hit my favorite wierdos: the Mobius strip, and the Klein bottle.

mobius_strip.jpg

We know that a cylinder is a product space - the product of a finite line segment and a circle. A Mobius strip is just a cylinder with a twist. So if we try to represent it as a fiber bundle, we'll find that it's a fiber bundle where the base space is a circle, and the fiber is a line segment. Looking at a Mobius strip, you can easily see the circular base; the base of a fiber bundle is generally pretty clear. The fiber is the part that can be twisted - and looking at a Mobius strip, the twisted fiber is clear. At any particular point, or any simple segment of the strip, it's obviously homoemorphic to a similar segment of a cylinder. So any neighborhood obviously looks like a cylindrical product space. But it's got that nasty twist in it, which means we can't build a proper projection function for the fiber, so it's not a product. But it does have most of the properties of a product, at least with respect to the circular base.

KleinGlass.jpg

The Klein bottle is very similar to the Mobius strip, only it's a lot harder to visualize - which is pretty natural since a true Klein bottle can't exist in three dimensions A Klein bottle is a fiber bundle with a circular base space, and a circular fiber space - the twisted version of a Torus. Again, looking at a figure of a Klein bottle, you can (if you stretch a bit) see the circular base. And looking at it, you should also be able to see that any finite section of it is homeomorphic to a cylinder - just like a section of a torus. But when you look at the full closed loop of the bottle, it's got that bizarre twist in it - so again, because of the twist, it can't be a product space; but it's got a clean projection to the base, and a clear homeomorphism to the toroidal product space for neighborhoods. So it's a very nice fiber bundle.

17 responses so far

  • The best characterization I ever heard, though I forget the ultimate source, works better face-to-face, but I'll try to explain here.
    A fiber bundle is something where the algebra goes this way (moves pointing finger up and down) and the topology goes this way (moves open hand, palm down, in a circle like polishing a table).

  • Mark C. Chu-Carroll says:

    John:
    That's a pretty good one. I've always had this image of putting a perfectly good topological space into a taffy-puller to twist it all around - but the way that it pulls and twists
    can't completely break it - it can just twist it up and wrap it around itself. It works for me, but I've never been able to get anyone else to see it ūüôā

  • Blake Stacey says:

    I thought the cross between a product and a manifold would be (product) (manifold) sin θ.

  • Mark C. Chu-Carroll says:

    Blake:
    Groan.

  • Where you say:

    For the projection function to work property, there must be an open neighborhood N including p where ŌÄ-1 is homeomorphic to N √ó F and the projection function from N √ó F to N correspond.

    I think you meant:

    For the projection function to work property, there must be an open neighborhood N including p where ŌÄ-1(N) is homeomorphic to N √ó F and the projection function from N √ó F to N correspond.

  • Torbj√∂rn Larsson says:

    Please, the math is hard enough without such obtuse technical math jargon as taffy pullers. It is hard to get that to commute... eh, compute.
    Speaking of bizarre twists, it seems to me the Möbius strip twists in one extra dimension (that it can be embedded in, resulting in changing up-down for a one-dimensional described line segment making a strip) while the Klein bottle twists in two (resulting in changing in-out for a two-dimensional described circle making a cylinder). Is that a correct intuition?

  • Mark C. Chu-Carroll says:

    Torbjörn:
    You got the Klein bottle just right - that's exactly what's going on.

  • Well, as long as we've got our levity working...
    I think "Fiber Bundles" would make an excellent breakfast cereal name. It's not the first one based on mathematics or computer science, though. That honor would go to the one named after the functional programming language for a quantum computer: "Quisp".

  • Torbj√∂rn Larsson says:

    Mark:
    Thank you! It is even difficult to ask properly when one doesn't have the foggiest idea of how to mathematically describe twists and embedding dimensions in this situation.

  • Anonymous says:

    It's very simple, Torbjörn - all you have to understand is that topology is for sissies, and real men only do algebra.

  • Alon Levy says:

    That was me in the last comment. I have no idea why the name was changed to Anonymous.

  • Alon: you might want to take that up with Atiyah, who notes that there's no such thing as analysis -- only algebra and geometry -- and that algebra is only there to talk about geometry.

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  • Torbj√∂rn Larsson says:

    Well, my first longer relationship was with analysis, and I still love to court exciting dynamical systems and nice distributions. Since the later are good euphemisms for women, I will still consider myself a manly man. If algebra is the heart of analysis I am perhaps not really mature enough to look much behind the alluring surface for such qualities.
    Topology is fun, but at times I will be tied up in knots.

  • Yin-Feng, Wang says:

    Dear friend, I hope that you can allow me to use the picture of the Klein bottle in my paper. So I need an email of permission from you.

  • juan says:

    Please see some more 3d plane-schematics of
    http://commons.wikimedia.org/wiki/Category:3-manifolds
    and are check what are fiber bunldes

  • Xezlec says:

    This elegant, intuitive explanation saved my sanity! Now I *finally* know what a fiber bundle is.

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