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Given nontrivial one-dimensional vector bundle $E$ on circle (which is of course Möbius strip). I need to find out if $E \oplus E \oplus ... \oplus E$ (n times) would be trivial or not (my guess is that it never happens) and the same question about $E \otimes E \otimes ... \otimes E$.

In the second case I can first of all notice that $E \otimes E$ is also one-dimensional and then we can look at transition maps. Let it be just multiplying by $-1$ (trivialization is considered in stereographic projections charts on circle). Well, then in $E \otimes E$ it will be $(-1) \otimes (-1) = 1 \otimes 1$ which makes this bundle into trivial one. For $E \otimes E \otimes E$ we obtain $-1 \otimes 1 \otimes 1= - id$ and so this bundle is not trivial and so on. We will obtain cylinder for even $n$ and Möbius strip for odd. Am I right?

I would appreciate some explanation regarding to first question and indication of mistakes (if there are) in my solution.

Invincible
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1 Answers1

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Real line bundles $L \to B$ are classified by their first Stiefel-Whitney class $w_1(L) \in H^1(L; \mathbb{Z}_2)$, in particular, $w_1(L) = 0$ if and only if $L$ is trivial. Therefore, if $E$ denotes the non-trivial line bundle on $S^1$, we see that $w_1(E)$ is the unique non-zero element of $H^1(S^1; \mathbb{Z}_2) \cong \mathbb{Z}_2$.

For line bundles $L_1$ and $L_2$, we have $w_1(L_1\oplus L_2) = w_1(L_1\otimes L_2) = w_1(L_1) + w_1(L_2)$, so for any line bundle $L$ we have

$$w_1(L^{\oplus n}) = w_1(L^{\otimes n}) = \begin{cases} 0 & n\ \text{even}\\ w_1(L) & n\ \text{odd}. \end{cases}$$

There are two cases:

  • if $L$ is trivial, then so is $L^{\oplus n}$ and $L^{\otimes n}$ for every $n$.
  • if $L$ is non-trivial, then $L^{\oplus n}$ is trivial if and only if $n$ is even, and $L^{\otimes n}$ is trivial if and only if $n$ is even.

In particular, as $E \to S^1$ is non-trivial, we are in the second case.

Michael Albanese
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