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I got very confused with understanding this theorem. So $\{y\}$ is a point, how could it be transversed by $f$?

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Proof: Given any $y \in Y.$ alter $f$ homotopically to make it transversal to $\{y\}$.

Thank you for your help~~

1LiterTears
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1 Answers1

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The tangent space to a point is trivial, so to say that $f$ is transverse to $\{y\}$ is just to say that $y$ is a regular value of $f$, i.e. that $f'(x):T_xX\to T_yY$ is surjective for all $x\in f^{-1}(y)$.

Zev Chonoles
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  • Aha! So that's just the image of $df_x$ needs to equal to $T_yY$. That is to say ${y}$ is a regular value! Thanks a lot Zev! – 1LiterTears Jul 13 '13 at 16:49
  • No problem, glad to help :) – Zev Chonoles Jul 13 '13 at 16:52
  • So I guess when $f$ does not pass through $y$, it is transversal and $I_2(f,{y}) = 0$; when $f$ pass through $y$, it is transversal when $y$ is a regular value, and $I_2(f,{y}) = 1$ right? Then, how come $I_2(f,{y})$ is the same for all $y \in Y$? (I'm happy to open a new question for this if you think that is better, and thanks in advance!) – 1LiterTears Jul 13 '13 at 17:17
  • Could you explain what you mean by $I_2$? – Zev Chonoles Jul 13 '13 at 17:20
  • Thanks a lot Zev! $I_2(f,Z)$ is the mod 2 intersection number of the map $f$. – 1LiterTears Jul 13 '13 at 17:23
  • Oh, it is the number of points $f^{-1}(Z)$ modulo 2. – 1LiterTears Jul 13 '13 at 17:36
  • Then it makes sense to me.. But why it has to do with transversality - just use the strong condition of transversality to show $y$ is a regular value...? – 1LiterTears Jul 13 '13 at 17:42
  • @Jellyfish: It's the same for all $y$ because $Y$ is connected and the number is locally constant by the Stack of Records Theorem. G&P use the transversality language so that this is defined even when $y$ is not a regular value by making a small wiggle of the map. – Ted Shifrin Jul 13 '13 at 18:46
  • Oh, got it. I know the locally constant by the stack of records theorem implies globally constant for connected space, but thanks so much @TedShifrin for clearing that $y$ can be wiggled into transversality. =) – 1LiterTears Jul 13 '13 at 19:52