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The theorem and its proof is given below:

enter image description here

But I could not understand the last line in the proof in particular:

Why $F^{-1}(Z)$ is a compact one dimensional manifold with boundary? And why this leads to

(#$\partial F^{-1}(Z)$ = #$f^{-1}(Z))$?

Could anyone explain this for me please?

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  • $\begingroup$ $F^{-1}(Z)$ is a manifold because of transversality. $\endgroup$ – Lord Shark the Unknown Dec 9 '18 at 3:48
  • $\begingroup$ By which theorem and why it is one dimensional?@LordSharktheUnknown $\endgroup$ – hopefully Dec 9 '18 at 3:57

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