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I have read in some texts (e.g. in https://www.math.uni-duesseldorf.de/~grk2240/pdf/ModuliVectorBundles.pdf) that complex vector bundles over a compact surface $S$ (usually a Riemann surface) are topologically characterized by their degree and rank. For rank one, i.e. line bundles, this is something I am familiar with. Does anybody have a reference for a proof for rank >1?

Another question: what about the smooth structure? Is it uniquely determined by the topology?

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If $E$ is a rank $r$ complex vector bundle over a CW complex $X$ of dimension $d$ with $2r > d$, then $E\cong E_0\oplus\varepsilon^1$ where $E_0$ is a complex vector bundle of rank $r - 1$ and $\varepsilon^1$ denotes the trivial rank one complex vector bundle, see this question. If $d = 2$, we can apply the above repeatedly to obtain $E \cong L\oplus\varepsilon^{r-1}$ for some complex line bundle $L$.

If $E'$ is another rank $r$ complex vector bundle, then $E' \cong L'\oplus\varepsilon^{r-1}$. Note that $c_1(E) = c_1(L)$ and $c_1(E') = c_1(L)$. Since complex line bundles are determined up to isomorphism by their first Chern class, it follows that $E \cong E'$ if and only if $c_1(E) = c_1(E')$. That is, complex vector bundles over a two-dimensional CW complex are determined up to isomorphism if and only if they have the same rank and first Chern class. Moreover, every possible choice of rank and first Chern classes arises.

In the case that $E \to S$ is a holomorphic vector bundle over a compact connected Riemann surface $S$, one can calculate the degree of $E$ topologically, namely $\deg E = \langle c_1(E), [S]\rangle$. Since $S$ is closed and oriented, the map $H^2(S; \mathbb{Z}) \to \mathbb{Z}$ given by $\alpha \mapsto \langle\alpha, [S]\rangle$ is an isomorphism. In particular, if $E'$ is another holomorphic vector bundle, then $\deg E = \deg E'$ if and only if $c_1(E) = c_1(E')$. So $E$ and $E'$ are isomorphic as complex vector bundles if and only if they have the same rank and degree.

As for your second question, a topological vector bundle over a smooth manifold admits a smooth vector bundle structure which is unique up to isomorphism.


More generally, complex vector bundles over a CW complex of dimension $\leq 4$ are determined up to isomorphism by their rank and Chern classes. Moreover, every possible choice of Chern classes and rank arises. See this answer for a proof.

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