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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 $S^4$?

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  • $\begingroup$ For the point singularity case, there is an example. Since outside the equator $S^3$, $S^4\setminus S^3$ can shrink to points,we can use the standard identification, but near S3S3 we use the coordinate of the point by $(\alpha,\beta)\to\rho(\sqrt{|\alpha|^2+|\beta|^2})\left(\begin{array}{cc}\alpha&-\bar\beta\\ \beta&\bar\alpha \end{array}\right)$, where we use the complex coordinate $(\alpha,\beta)$ and $\rho$ is the cut off function near the point, by the partition of unity, we can have a global mapping. But I do not how to give the sphere singularity example. $\endgroup$ – DLIN Jul 22 '16 at 11:20
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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 $v=f\cdot\text{id}$.

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  • $\begingroup$ Could you give the reference of that Lemma. Well, I want to find two different vector bundles $E_0,E_1$. But is is also usefull. Thank you for your sharing.@nsmath $\endgroup$ – DLIN Jul 26 '16 at 3:44
  • $\begingroup$ @DLIN For example, see the Theorem 2.29 in a Lee's textbook Introduction to Smooth Manifolds Second Edition (available free online). Be careful of the Edition number. $\endgroup$ – Shinichiro Nakamura Jul 26 '16 at 8:07

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