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I am reading Lee's Introduction to Smooth Manifolds and got stuck on the problem 5.6. The question is written here, question 1. (There is a typo in the question. The last sentence should be "Show that F is smoothly isomorphic...")

I do not know how to use the transition function in order to show that these bundles are isomorphic. I tried to find smooth trivializations of F, but I could not.

Edit: This is how Lee defined the Möbius bundle in his book. (Lee, page 105, example 5.2)

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It is more or less impossible to answer without know how the Moebius bundle was defined! –  Mariano Suárez-Alvarez Dec 4 '11 at 1:15

2 Answers 2

The bundle chart neighborhoods were given to you in the exercise setup, $U$ and $V$. To define the smooth trivializations of $F$, remember that you want a real line bundle, so the fiber is $\mathbb{R}$, and the neighborhoods are called trivializations for a reason. With the transition functions $\tau$, can construct $F$ (it is a theorem you have probably seen in your class that transition functions with certain conditions + trivializations totally characterize a vector bundle). You are set after taking the maximal bundle atlas for $F$ generated by your charts.

Now you need to construct an isomorphism between $F$ and the Möbius band. This is so painfully simple it's hard. I can't give details since you haven't told us how your professor defined the Möbius strip in class, but you'll probably need to construct maps from $U$ and $V$ into the Möbius band, show that precomposition by $\tau$ transforms them to one another, so that you have a well-defined map of fiber bundles.

Then you're done with the hard work, but you will probably still need to chase definitions to show that this is an isomorphism on each fiber, it covers a homeomorphism of $S^1$, and so forth.

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$F$ and the Möbius bundle $M$ are both rank-1 vector bundles over $S^1$. Let {$\tau_{ab}$} and {$\tilde{\tau_{ab}}$} denote the transition functions determined by the local trivializations of $F$ and $M$, respectively. Then $F$ and $M$ are smoothly isomorphic over $S^1$ iff for each $U_\alpha$ in the open cover of $S^1$, there exists a smooth map $\sigma_\alpha: U_\alpha \rightarrow GL(1,\mathbb{R})$ such that $$\tilde{\tau_{ab}}(p)= \sigma_\alpha(p)^{-1}\tau_{ab}(p)\sigma_\beta(p) $$

where $p$ is in the intersection of $U_\alpha$ and $U_\beta$.

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