A cubic surface, fibered by elliptic curves.pic.twitter.com/fbKhRYO9ef
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Here it’s like a fiber bundle (which is like a vector bundle but for more general types of spaces) except there can be some “singularities” or points where some of the spaces are bad. That happens three times in this example - when circles break off / recombine or collapse.
i see .. i still am not sure what a Fiber bundle is with 100% confidence .. wanna ELi5 for good measure (no pun intended)?
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For sure! Say you got a bunch (maybe...a bundle?) of sticks or wheat all tied together, and you want to describe to me. You could break the description into two pieces: 1) the shape of a piece of wheat, and 2) the shape of the cross section of your bundlepic.twitter.com/d46uiwuciM
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By telling me you've stacked a bunch of wheat together so the cross section looks like a disk, I could develop a sort of picture for what you are talking about (the bundle of wheat, a 3d thing) in terms of simpler data: a single fiber of wheat (1d) and the cross section (2d)
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A level up: if you have a map f:X->Y, then the subsets f^{-1}(y) are called fibers. Sometimes "the fiber over y" The first thing to know is a fiber bundle is a map where all the fibers are the same - no matter what y is, we get the same thing as the fiber over y.
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The first example is the "trivial bundle" Y=XxF, and the map f:Y->X is just projection onto the first coordinate f(a,b)=a. The other condition we put on a fiber bundle is that locally, it looks like the trivial bundle.
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