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Could someone give me some help to get started with this question? Don't even have the slightest idea.. =(

Suppose a vector field v on $\mathbb{R}^n$ has exactly two isolated zeros $p, q$, and $p, q$ are connected by a flow-line of the vector field. Furthermore, assume one can modify the vector field in a compact neighborhood of the flow line and merge $p, q$ to give new vector field $w$ with a single isolated zero $r$. Show that the index of $r$ must be the sum of the indices of $p$ and $q$.

Thank you.

WishingFish
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  • What is the definition of your "index"? Have you worked with the Hopf-Index theorem? – Chris Gerig Jul 03 '13 at 23:59
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    This requires a good deal of machinery. Your earlier questions about differential topology were at the beginning of the subject. How did you advance through three months of material to this? – Ted Shifrin Jul 03 '13 at 23:59
  • @TedShifrin Oh I am just very unfortunate to encounter this question... – WishingFish Jul 04 '13 at 00:13
  • @ChrisGerig According to Guillemin & Pallock, the index of $\vec{v}$ at 0, to be the degree of this directional map $S_\epsilon \rightarrow S^{k-1}$. – WishingFish Jul 04 '13 at 00:16
  • You need a version of the Boundary Theorem, which will give you (generalizing the Argument Principle in complex analysis) such a conservation law, once you have all the definitions and tools. – Ted Shifrin Jul 04 '13 at 01:17
  • Oh Thanks a lot @TedShifrin. That will take me sometime! =) – WishingFish Jul 04 '13 at 01:19
  • @TedShifrin what is the Boundary theorem you are referring to? – Luigi M Aug 22 '17 at 14:24
  • @LuigiM: The Boundary Theorem says that if $M$ is a compact orientable $(n+1)$-dimensional manifold with boundary and $N$ is a compact orientable $n$-dimensional manifold, then a map $f\colon\partial M\to N$ extends to $M$ if and only if $\deg(f)=0$. – Ted Shifrin Aug 22 '17 at 23:51

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