Why does this short exact sequence split?

Question

I am currently reading K-Theory, Anderson and Atiyah, and trying to understand the index bundle (of families of Fredholm operators).

Here is some context: let $X$ be a compact topological space, $F\subseteq B(H)$ be the set of bounded Fredholm operators in a Hilbert space $H$ equipped with the norm topology induced by $B(H)$, and $T:X\to F$ be a continuous map. There exists a closed subspace $V$ of $H$ of finite codimension such that $V\cap\ker T_x = 0$ for all $x\in X$, and it was proved that $H/T(V)=\bigsqcup\limits_{x\in X} H/T_x(V)$ is a vector bundle over $X$ under such conditions. One defines $$\mbox{Ind}(T) = [H/V]-[H/T(V)] \in K(X),$$ where $H/V$ stands for the trivial bundle $X\times H/V$.

In order to prove that this definition does not depend on $V$, it suffices to prove that if $W\subseteq V$ is another closed subspace of $H$ of finite codimension, then $$[H/W]-[H/T(W)] = [H/V]-[H/T(V)]$$ in $K(X)$.

Now, what I think Atiyah done in page 158 is:

Statement: The existente of the short exact sequence of vector bundles $$0\longrightarrow V/W \longrightarrow H/T(W) \longrightarrow H/T(V) \longrightarrow 0$$ implies immediately that $[H/T(W)] = [H/T(V)] + [V/W]$ in $K(X)$.

I can't see the reason why this is true.

I tried to prove that this short exact sequence splits, unsuccessfully. Here is what I did:

1. I have showed that $T(V)/T(W) = \bigsqcup\limits_{x\in X} T_x(V)/T_x(W)$ is a vector bundle isomorphic to the trivial bundle $V/W$. Then the sequence becomes simply $$0 \longrightarrow T(V)/T(W) \overset{i}{\longrightarrow} H/T(W) \longrightarrow H/T(V) \longrightarrow 0$$ where $i$ is induced by the inclusions $T_x(V)\subseteq H$.

2. Next, I tried to split the map $i$, constructing the map $p:H/T(W)\to T(V)/T(W)$ given by $p_x(u_x+T_x(W)) = P_x(u_x)+T_x(W)$, where $P_x:H\to H$ is the orthogonal projection onto $T_x(V)$. I showed that $p$ is well defined, but I'm having trouble to prove continuity of $p$. In fact, it easy to see that it suffices to show that the map $X\times H\to H$ given by $(x,u)\mapsto P_x(u)$ is continuous. Here is what I tried in this sense, without success.

Question: How can I prove this statement by Atiyah ?

Am I in the right direction? Any help is appreciate. Thanks in advance.

Answer 1

It seems to me that you are trying to work at the level of vector bundles with infinite rank, which certainly complicates things. However, these quotients are all finite rank, so we can apply the following:

If $X$ is a paracompact space, then every short exact sequence $$0 \to A \to B \to C \to 0$$ of (finite rank) vector bundles over $X$ can be split.

Indeed, since $X$ is paracompact, we can always equip $B$ with a metric. This allows us to take orthogonal projection $B \to A$, and further allows us to identify $C$ with the orthogonal complement $A^\perp$ of $A$.

More precise details can be found in section I.4 in Anderson-Atiyah.