- #1

Ghost Repeater

- 32

- 5

Here's the text:

Then he says (and this is the bit where he loses me):"Let E be a fibre bundle E → S^{1}with a typical fibre F = [-1, 1]. Let U_{1}= (0, 2π) and U_{2}= (-π,π) be an open covering of S^{1}and let A = (0,π) and B = (-π,π) be the intersection U_{1}∩ U_{2}. The local trivializations φ_{1}and φ_{2}are given by

φ_{1}^{-1}(u) = (θ, t), φ_{2}^{-1}(u) = (θ, t)

for θ ∈ A and t ∈ F. The transition function t_{12}(θ), θ ∈ A, is the identity map t_{12}(θ): t → t."

My questions are:'We have two choices on B:

I) φ_{1}^{-1}(u) = (θ, t), φ_{2}^{-1}(u)=(θ, t)

II) φ_{1}^{-1}(u)=(θ, t), φ_{2}^{-1}(u) = (θ, -t)"

i) The construction of the problem confuses me a little, because how can two open sets A and B make an intersection? Why did Nakahara take the intersection of U1 and U2 and split it up this way?

ii) How do we know that we have these two choices on B? It seems like it's supposed to be self-evident, but it isn't to me. Why do we have these two choices on B but not on A?