I'm reading the construction of difference bundle(generalized) in the paper of Atiyah-Hilzebruch(Analytic cycles on complex manifolds)

There is one thing I cannot understand. The followings are in the page 34 in the paper.

For exact sequence of vector bundles on $Y$.

$0 \to E_n \stackrel{a_n}{\to} E_{n-1} \to \dots \to E_1 \stackrel{a_1}{\to} E_0 \stackrel{a_0}{\to} 0$

Let $F_i := Ker(a_i)$. Then $F_i$ are vector bundles and the following short exact sequences

$0 \to F_i \to E_i \to F_{i-1} \to 0$ splits.

I cannot find the reason why they splits and there is no condition on the space $Y$. Is there a reason the above sequence splits, or the term 'split' used in a different sense?

I'm reading the construction of difference bundle(generalized) in the paper of Atiyah-Hirzebruch(Analytic cycles on complex manifolds)

There is one thing I cannot understand. The followings are in the page 34 in the paper.

For exact sequence of vector bundles on $Y$.

$0 \to E_n \stackrel{a_n}{\to} E_{n-1} \to \dots \to E_1 \stackrel{a_1}{\to} E_0 \stackrel{a_0}{\to} 0$

Let $F_i := Ker(a_i)$. Then $F_i$ are vector bundles and the following short exact sequences

$0 \to F_i \to E_i \to F_{i-1} \to 0$ splits.

I cannot find the reason why they splits and there is no condition on the space $Y$. Is there a reason the above sequence splits, or the term 'split' used in a different sense?

I'm reading the construction of difference bundle(generalized) in the paper of Atiyah-Hilzebruch(Analytic cycles on complex manifolds)

There is one thing I cannot understand. The followings are in the page 34 in the paper.

For exact sequence of vector bundles on $Y$.

$0 \to E_n \stackrel{a_n}{\to} E_{n-1} \to \dots \to E_1 \stackrel{a_1}{\to} E_0 \stackrel{a_0}{\to} 0$

Let $F_i := Ker(a_i)$. Then $F_i$ are vector bundles and the following short exact sequences

$0 \to F_i \to E_i \to F_{i-1} \to 0$ splits.

I cannot find the reason why they splits and there is no condition on the space $Y$. Is there a reason the above sequence splits, or the term 'split' used in a differenct sence?

I'm reading the construction of difference bundle(generalized) in the paper of Atiyah-Hilzebruch(Analytic cycles on complex manifolds)

There is one thing I cannot understand. The followings are in the page 34 in the paper.

For exact sequence of vector bundles on $Y$.

$0 \to E_n \stackrel{a_n}{\to} E_{n-1} \to \dots \to E_1 \stackrel{a_1}{\to} E_0 \stackrel{a_0}{\to} 0$

Let $F_i := Ker(a_i)$. Then $F_i$ are vector bundles and the following short exact sequences

$0 \to F_i \to E_i \to F_{i-1} \to 0$ splits.

I cannot find the reason why they splits and there is no condition on the space $Y$. Is there a reason the above sequence splits, or the term 'split' used in a different sense?