{"id":291,"date":"2007-01-28T10:32:03","date_gmt":"2007-01-28T10:32:03","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2007\/01\/28\/twisted-spaces-fiber-bundles\/"},"modified":"2007-01-28T10:32:03","modified_gmt":"2007-01-28T10:32:03","slug":"twisted-spaces-fiber-bundles","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2007\/01\/28\/twisted-spaces-fiber-bundles\/","title":{"rendered":"Twisted Spaces: Fiber Bundles"},"content":{"rendered":"

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<\/em>. 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.)<\/p>\n

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<\/em> to be euclidean, but the space as a whole may be very non-<\/em>euclidean. There are all sorts of interesting things that you can do in a manifold because of that property of being locally<\/em> almost-euclidean – things like calculus.<\/p>\n

A fiber bundle is based on a similar sort of idea: a local<\/em> property that does not<\/em> necessarily hold globally – but instead the local property being a property of individual points, it’s based on a property of regions<\/em> of the space.<\/p>\n

So what is a fiber bundle, and why should we care? It’s something that looks almost<\/em> 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.<\/p>\n

<\/p>\n

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<\/em> a product space – it’s the product of a line and a circle; the mobius strip is not<\/em> a product space, because it’s got a twist in it. But in most<\/em> respects, it behaves<\/em> 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. <\/p>\n

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<\/em>), E (the total space<\/em>, and F (the Fiber<\/em>); and the mapping function is called π (the projection map<\/em>). π : E → B is a continuous onto<\/em> 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.) <\/p>\n

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.<\/p>\n

Here’s the tricky part. Suppose we have a point p<\/em> in B. For the projection function to work property, there must be an open neighborhood N including p where π-1<\/sup> 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<\/em> is a homeomorphism from π-1<\/sup>(N) to N×F, and pN<\/sub> is the projection function from N×F to N. The requirement on the fiber bundle’s projection function is that the following diagram commutes:<\/p>\n


\n\"bundle-cat.jpg\"<\/p>\n

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. <\/p>\n

\"mobius_strip.jpg\"<\/p>\n

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<\/em> a product. But it does have most of the properties of a product, at least with respect to the circular base.<\/p>\n

\"KleinGlass.jpg\"<\/p>\n

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<\/em> 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.<\/p>\n","protected":false},"excerpt":{"rendered":"

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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[65],"tags":[],"class_list":["post-291","post","type-post","status-publish","format-standard","hentry","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-4H","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/291","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=291"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/291\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=291"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=291"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=291"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}