As I understand it, an open cover of a Base Space and associated holomorphic transition functions on the intersection are sufficient data to define (up to isomorphism perhaps) a holomorphic (complex) line bundle.
So if we cover S1 with open sets U and V which intersect in open sets P and Q such that P and Q have empty intersection (you will note I am avoiding Latex here. Basically we cover S1 with a pair of horseshoes). We define the UV- transition function to be +1 on P and -1 on Q (locally constant thus holomorphic) we have the set up for an (infinite) Mobius band.
What I am strugling to define are the corresponding trivialisations from U or V to UxC or VxC
that correspond to this transition function.
Even Grifiths & Harris has some "hand waving" about identifying points which does not help. I just want to see the explicit map on the fibres over U and V.


Any suggestions?


Thanks
Noel R



Trivialisation of Mobius Line Bundle
There appears to be some confusion in the question: The circle $S^1$ is not a complex manifold, so it does not admit a meaningful notion of holomorphic line bundle. If you try to construct a complex line bundle on the circle, you will find that it is automatically a trivial line bundle. If you want to construct a Möbius band as a line bundle, you should take a real line bundle, with the transition functions you specified.


I would recommend a more topological text than Griffiths and Harris for information about the construction you are trying to do. Perhaps Hatcher's book on vector bundles and K-Theory will help. It is available on his web page.


