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If $M$ is paracompact (let us assume smooth manifold of dimension $d$), then one has that the set of isomorphism classes of vector bundles of rank $n$ over $M$ is isomorphic to the set of homotopy classes $V_n:=[M,G_n\left(\mathbb{R}^{n+d}\right)]$ from $M$ to a Grassmannian. This holds both in the topological and the smooth category over $M$.

I was wondering if there is any "nice" topological/smooth structure on $V_n$?

Does it carry a well-behaved and non-trivial topology?

Are there results on how "many" elements $V_n$ usually has? Finitely many, Countably many?

This is really not my area, so these questions may have simple, well-known answers.

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    $\begingroup$ You've taken homotopy classes so the result is just a set. For reasonable $M$ it should just be countable. Are you asking what happens if you don't take homotopy classes? $\endgroup$
    – Qiaochu Yuan
    Jan 11 at 18:43
  • $\begingroup$ @QiaochuYuan My question was/is does the set of homotopy classes of such maps carry an interesting topological or smooth structure. Do you have have "reasonable" conditions in mind for countability? If the set is countable then the question for smooth/topological structure is of course rather moot. The question arose, because i was wondering if the moduli space of smooth vector bundles is a smooth manifold. Similar to some moduli spaces in complex geometry for example. $\endgroup$
    – t312t
    Jan 11 at 20:38
  • $\begingroup$ The "moduli space of vector bundles," depending on what exactly you mean by this, can be thought of as the entire space of maps from $M$ to a classifying space $BGL_n$, rather than just the set of homotopy classes of such maps. This space has an interesting homotopy type and its $\pi_0$ is the set of isomorphism classes which is just a set (and should be countable if, say, $M$ is compact, but I'm not sure about this). In the smooth category there is a version of this object which is a smooth stack and so which carries smooth structure in that sense. $\endgroup$
    – Qiaochu Yuan
    Jan 11 at 21:28
  • $\begingroup$ But at the most basic level, in the topological and smooth categories vector bundles only have discrete invariants (e.g. characteristic classes) and only vary discretely; they do not vary continuously the way objects in complex and algebraic geometry do. $\endgroup$
    – Qiaochu Yuan
    Jan 11 at 21:30

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