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I would like to review the definition of the degree (Brouwer and mod.2) of an application between smooth manifolds. Let $f:M\to N$ be a smooth map:

  1. Is the set $R$ of regular values open and dense in $N$? or just a countable intersection of open dense subsets? (as asserted in Milnor's book, Topology from Differentiable Viewpoint, Brown's Theorem page 11)
  2. If $M$ is compact, $N$ connected and $\dim M=\dim N$. Is the number of elements of the fibre $f^{1}(y)$ constant in $N$? ($R$ is not necessary connected if $N$ is!)
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  • $\begingroup$ 1) Yes. Prove this. 2) No. Take, say, the identity map $S^1 \to S^1$ and perturb it a little bit so that $f^{-1}(1)$ has three points, $f^{-1}(-1)$ has one. Note that when you do this, $f$ will not be a submersion. If it is a submersion, then indeed $|f^{-1}(y)|$ is constant, because $f$ is a covering map. $\endgroup$ – user98602 Nov 12 '15 at 19:54
  • $\begingroup$ 2. Theorem: That number is constant mod 2. That is if you take two different $y$'s, the difference in the number of their respective pre-images can only differ by an even number. $\endgroup$ – Behnam Esmayli Aug 13 '16 at 18:04

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