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Rasalhague

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*The Road to Reality*, § 15.2, Roger Penrose introduces the concept of a fibre bundle. Here I'll modify his notation to that of Wikipedia and other sources, so that B stands for base space (rather than the bundle), F for fibre, and E for the total space, which I think Penrose calls the (fibre) bundle itself.

His first example is the real line bundle over S

^{1}. This is the product space S

^{1}x R

^{1}, where S

^{1}is a circle (any circle?) and R

^{1}is the 1-dimensional vector space of real numbers over the reals with the natural definition of scalar multiplication. A product space is called a trivial bundle of F over B. So far so good. 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 [itex]p \; \circ \; s(x) = x[/itex]. (This according to Wikipedia's more formal intro; Penrose's cross-section is only the image of this map, or can be "thought of as" the image; he's a little vague here.) At least, I think can see how this works for a product B x F.

But how is it different in the twisted case? Penrose, having just described the case for a trivial bundle B x F: "This is like the ordinary idea of the

*graph*of a function [...] More generally, for a twisted bundle B, any cross-section of B defines a notion of a 'twisted function' that is more general than the ordinary idea of a function." In Fig. 15.7, he shows a curve drawn in a loop around a cylinder, and one around a Möbius strip. I think the idea is that the loop can't go all the way around the Möbius strip

*and join up with itself*(and thus can't be continuous) unless it crosses the central line around the strip (which line Penrose identifies with the zero section).

I suppose what's puzzling me is that these copies of F are disjoint, so how can one copy of, say, the vector space (R

^{1}, R

^{1}, regular multiplication) be said to have something analogous to an orientation or direction compared to another? Whereabouts in the definition of the Möbius strip bundle does this idea come in?