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John Klein
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The following is basically thoroughly usless general nonsense. It's main problem is that it lacks geometry. Even so, it kind of does the job and it is probably worth mentioning.

Let $G$ be the structure group of your vector bundle. I will assume that $G$ is connected (for example, $G$ could be $U(n)$ or $SO(n)$). Let our vector bundle be classified by a map $f\: X\to BG$.

Definition: Let $k \ge 1$ be an integer. A $k$-structure is a pair $(Z,g)$ such that

$\bullet$ $Z$ is a path connected space.

$\bullet$ $Z$ has vanishing homotopy groups above dimension $k$, and

$\bullet$ $g: Z\to BG\, $ is a map.

[To such pairs $(Z_i,g_i)$ for $i=0,1$ and are equivalent there is a (finite chain of) weak homotopy equivalences over $BG$ from $Z_0$ to $Z_1$.]

Example: A 1-structure amounts to a discrete group $H$ and a map $BH\to BG$.

Definition: A $k$-flat reduction of $f$ consists of a $k$-structure $(Z,g)$ and a factorization of $f$ up to homotopy as $\require{AMScd}$ $$ \begin{CD} X \to Z @>g>> BG\, . \\ \end{CD} $$

Examples:

(1). A 1-flat reduction of $f$ is the same thing as a flat reduction of the associated vector bundle.

(2). An oriented line bundle over $X$ automatically admits a $2$-flat reduction (this is a tautology, since $BSO(2) = K(\Bbb Z,2)$. In particular, the canonical line bundle over $\Bbb CP^1$ admits a 2-flat reduction but not a 1-flat reduction (since it isn't flat). More generally, a finite Whitney sum of oriented line bundles admits a $2$-flat reduction, since $BSO(2)^{\times j} = K(\Bbb Z^j,2)$.

John Klein
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