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In Chapter 7, Lemma 3, Milnor writes:

Lemma. If $f$ and $g$ are smoothly homotopic and $y$ is a regular value for both, then $f^{-1}(y)$ is framed cobordant to $g^{-1}(y)$.
Proof: Choose a homotopy $F$ with $$F(x, t)=f(x), 0\leq t<\epsilon\\F(x, t)=g(x), 1-\epsilon<t\leq 1.$$ Choose a regular value $z$ for $F$ which is close enough to $y$ so that $f^{-1}(z)$ is framed cobordant to $f^{-1}(y)$ and so that $g^{-1}(z)$ is framed cobordant to $g^{-1}(y)$. Then $F^{-1}(z)$ is a framed manifold and provides a framed cobordism between $f^{-1}(z)$ and $g^{-1}(z)$. This proves the lemma.

I wonder why he didn't say directly that $F^{-1}(y)$ is a framed manifold and provides a framed cobordism between $f^{-1}(y)$ and $g^{-1}(y)$? Is it right to say this?

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  • $\begingroup$ OK, I know. It's because $y$ is not necessarily a regular value of $F$. $\endgroup$
    – wyhorgyh
    Commented Dec 14, 2023 at 7:46

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