In the last weeks I have been think of the transition from topological vector bundles to smooth and holomorphic vector bundles. This has resulted in a few questions (with a common thread) as follows: Always $\pi:E \to B$ is a topological (complex) vector bundle over a compact base,
(A) For any given smooth manifold structure on $B$, can there exist more than one differential structure on $E$ giving $\pi:E \to B$ the structure of a smooth vector bundle. If so, what is an example?
(B) Same question as above but replacing smooth by holomorphic.
(C) For a choice of smooth vector bundle structure on $\pi:E \to B$, does the de Rham complex of $E$ have an easy relationship with the de Rham complex of $B$. A (very) naive guess would be that
$$
\Omega^{\bullet}(E) = \Gamma^{\infty}(E) \otimes_{C^\infty(B)}\Omega^{\bullet}(B),
$$
but I can't see that there is a well-defined way to define the differential.
(D) Same question as above but for holomorphic structures and the Dolbeault complex
Best Answer
The answer to A is no. The topological bundle is determined by a continuous homotopy class of maps into the classifying space. Choosing a compatible smooth structure means picking a smooth map in that class. Any two such choices are smoothly homotopic. B however is true. Look up the Jacobian of a Riemann surface. How are you going to grade in C ? Where are the forms on $E$ of degree higher then the dimension of $B$?