At page 22, when proving the "Homotopy lemma", Milnor uses the fact that $|f^{-1}(y) + g^{-1}(y)| =|f^{-1}(y)| + |g^{-1}(y)|$, (thus he concludes that $0\equiv \partial F = |f^{-1}(y)| + |g^{-1}(y)|$, i.e. the thesis). I don't see why this should be true, what assures that $|f^{-1}(y) \cap g^{-1}(y)|$ is zero?
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5Page $22$ of what? What is $f$? what is $g$? Does $|\cdots |$ denote absolute value or cardinality? – lulu Jan 03 '23 at 18:52
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Your statement is a misrepresentation of what Milnor is doing. He has $F^{-1}(y)\cap M\times\{0\} = f^{-1}(y)\times \{0\}$ and $G^{-1}(y)\cap M\times \{1\} = g^{-1}(y)\times\{1\}$. The two sets are disjoint because of the different $t$ coordinates.
Ted Shifrin
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You’re welcome. You should accept the answer unless you have further questions. – Ted Shifrin Jan 05 '23 at 19:55