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The following is thoroughly useless general nonsense. Its 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. For simplicity, 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 $BSO(2)^{\times j} = K(\Bbb Z^j,2)$.

(3). I don't know of examples in degrees $\ge 2$. However, if one is willing to work instead in the rational homotopy category, there are examples in higher degrees. Here's one: let $X= S^4$. This includes into $BS^3 = BSU(2)$. Rationally, this is a $K(\Bbb Q,4)$. Taking $G = SU(2)$ and $f: X\to BG$ to be the inclusion (which represents the quaternionic line bundle), we see that $f$ is rationally $4$-flat but not rationally $3$-flat.

Correction: the definition below is wrong. It isn't true that a 1-flat reduction is the same as a flat reduction. One also needs to require that the map $Z \to BG$ factors through $BG^\delta$, where $G^\delta$ is $G$ with the discrete topology. (Also, one may as well replace $Z$ by the $k$-th Postnikov section in the definition of a $k$-structure).

The following is thoroughly useless general nonsense. Its 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. For simplicity, 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 $BSO(2)^{\times j} = K(\Bbb Z^j,2)$.

(3). I don't know of examples in degrees $\ge 2$. However, if one is willing to work instead in the rational homotopy category, there are examples in higher degrees. Here's one: let $X= S^4$. This includes into $BS^3 = BSU(2)$. Rationally, this is a $K(\Bbb Q,4)$. Taking $G = SU(2)$ and $f: X\to BG$ to be the inclusion (which represents the quaternionic line bundle), we see that $f$ is rationally $4$-flat but not rationally $3$-flat.

The following is thoroughly useless general nonsense. Its 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. For simplicity, 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 $BSO(2)^{\times j} = K(\Bbb Z^j,2)$.

The following is thoroughly useless general nonsense. Its 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. For simplicity, 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 $BSO(2)^{\times j} = K(\Bbb Z^j,2)$.

(3). I don't know of examples in degrees $\ge 2$. However, if one is willing to work instead in the rational homotopy category, there are examples in higher degrees. Here's one: let $X= S^4$. This includes into $BS^3 = BSU(2)$. Rationally, this is a $K(\Bbb Q,4)$. Taking $G = SU(2)$ and $f: X\to BG$ to be the inclusion (which represents the quaternionic line bundle), we see that $f$ is rationally $4$-flat but not rationally $3$-flat.

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 $BSO(2)^{\times j} = K(\Bbb Z^j,2)$.

The following is thoroughly useless general nonsense. Its 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. For simplicity, 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 $BSO(2)^{\times j} = K(\Bbb Z^j,2)$.