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Let $\xi$ be a $O(n)$-bundle with fibre $\mathbb{R}^n$. Let $\xi\otimes \mathbb{C}$, $\xi\otimes \mathbb{H}$ be complex vector bundles and quaternionic vector bundles.

If $\xi$ is not a trivial bundle, can we obtain $\xi\otimes \mathbb{C}$, $\xi\otimes \mathbb{H}$ are not trivial bundles?

Shiquan
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  • The question seems strangely phrased. You're asking if, given any non-trivial $O(n)$ bundle (over a space $X$), the complexification will remain non-trivial? In other words, is there a single non-trivial $O(n)$-bundle whose complexification is non-trivial? – Jonathan Beardsley Feb 10 '15 at 02:49
  • Or perhaps you're just asking whether or not it is always true that given a non-trivial bundle the complexification and quaternionification bundles are non-trivial? – Jonathan Beardsley Feb 10 '15 at 02:50

2 Answers2

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The Möbius real line bundle bundle $\xi$ over the circle $S^1$ is not trivial but its complexification $\xi\otimes_\mathbb R \mathbb C$ is trivial, like all complex line bundles over $S^1$.
[This last fact is due to complex line bundles on the circle being classified by $H^2(S^1,\mathbb Z)=0$]

Georges Elencwajg
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The tangent Bundle of sphere is another example. For arbitrary vector bundle E, it seems that $E\otimes \mathbb{C}$ has a trivial line bundle, as a summand.

https://mathoverflow.net/questions/209247/almost-complex-structure-and-nontrivial-idempotents

Moreover, I guess that the complexification of canonical line bundle over $\mathbb{R}P^{n}\;\;n>1$ is nontrivial.

Community
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Ali Taghavi
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