I want to construct a vector bundle homomorphism on some manifold $M$ whose singularity, set of points where is not isomorphism, is equal to an odd-dimensional sphere $S^{2k-1}$.
The following is just an idea.
$M:=S^4$, since $S^1\hookrightarrow S^4$, and its tubular neighborhood is trivial $N(S^1)=S^1\times D^3$. Let $E_1=\mathbb C^2\times M$, $E_2=\mathbb C^2\times N(S^1)\coprod \mathbb C^2\times (M\setminus N(S^1))/\sim$, where $\sim $ means on the boundary $\mathbb C^2\times S^1\times S^2$, $(x,v)\sim(x,g(x)v)$ for some $g:S^1\times S^2\to GL_2(\mathbb C)$.
$E_1\overset{v}{\to}E_2$ is by $v:=r\rho(r)g+(1-\rho(r))Id$, where $r$ means in a small neighborhood of $S^1$ the normal radius to $S^1$ and $\rho$ is just a cut-off function.
Q: I think the above map $v$ satisfies $Sing(v)=S^1$, is it smooth?