Let $\varphi:S^k \times D^l \to M$ be a smooth map of $(k+l)$-dimensional Manifolds. Let $\epsilon$ denote the trivial bundle.
In our lecture notes we showed that there is a stable trivialization of $T(S^k \times D^l)\oplus \varphi^*(TM)$ and therefore (?) there has to be a bundle map $T(S^k \times D^l)\oplus \epsilon^a\to TM\oplus \epsilon^a$ covering $\varphi$.
But why does this trivialization induce this bundle map? And how does this bundle map look like?
Do we have maps of the summands into the whitney sum? Then we could get the following map, which should be the desired map. $$T(S^k \times D^l)\oplus \epsilon^a \to T(S^k \times D^l)\oplus \varphi^*(TM)\oplus \epsilon^a = \epsilon^{a+b} \to \varphi^*(TM)\oplus \epsilon^a \to TM\oplus \epsilon^a$$ If not, how do I get this map?