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I know it is easy to classify line bundles over $\mathbb{R}P^2$. But do we have a classification of two dimensional complex vector bundles over $\mathbb{R}P^2$?

Zhaoting Wei
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    It follows from obstruction theory that for a complex vector bundle $E$ over a CW complex $X$ with $\operatorname{rank}{\mathbb{R}}E > \dim X$, there is a complex vector bundle $E_0$ with $\operatorname{rank}{\mathbb{R}}E_0 \leq \dim X$ such that $E \cong E_0 \oplus \varepsilon^d_{\mathbb{C}}$; see here for example. So every rank two complex vector bundle over $\mathbb{RP}^2$ is of the form $L\oplus\varepsilon^1_{\mathbb{C}}$ for some complex line bundle. As $H^2(\mathbb{RP}^2; \mathbb{Z}) \cong \mathbb{Z}_2$, there are only two such bundles. – Michael Albanese May 28 '22 at 10:11
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    More generally, complex vector bundles over a CW complex of dimension $\leq 4$ are classified by their Chern classes and rank. Moreover, every possible choice of Chern classes and rank arises. See here. – Michael Albanese May 28 '22 at 10:16

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