Could anyone give a non-trivial example of a bundle-mapping over $S^4$, i.e. find two complex rank 2 vector bundles $E\_0,E\_1$ over $S^4$ and a bundle mapping 
$$0\to E\_0\overset{v}{\to}E\_1\to0$$
such that the singularity set of $v$(where $v$ is not an isomorphic mapping) is equal to the embedding sphere $S^2$ of $S^4$?



Example of bundle-mapping over $S^4$ with singularity $S^2$
The answer is YES. First of all, notice the following lemma :



> 
> Let $M$ be a smooth manifold and let $K$ be a closed subset in $M$. Then, there exist a smooth function $f$ on $M$ such that $f^{-1}(0)=K$.
> 
> 
> 


We can deduce this lemma by using a partition of unity.


Then, for $M=S^4$ and $K=S^2$, you can construct such an example by setting $E\_0=E\_1=E$ (where $E$ is an arbitrary complex vector bundle) and $v=f\cdot\text{id}$.


