# Final step of homotopy lemma

In proving Homotopy lemma in Milnor Topology from the differential viewpoint we consider $V_1 \cap V_2$ where $V_1$ is a neighbourhood of $y$ in which $card f^{-1}( y)$ is constant. Similarly $V_2$ s a neighbourhood of $y$ in which $card g^{-1}( y)$ is constant. If $F$ is smooth homotopy between $f$ and $g$ then he chooses $z\in V_1\cap V_2$ such that $z$ is regular value of $F$. How do we know such $z$ exists ?

$V_1 \cap V_2$ is non-empty (since $y$ is in both terms) and open. If there were no regular values for $F$ there, you'd have a set of positive measure containing only critical values. This contradicts Sard. (See also Brown's corollary on page 11 of the Princeton Landmarks version.)

• Excuse me. Why did Milnor conclude that $\#f^{-1}(z)\equiv\#g^{-1}(z)$ in the last paragraph? – Steve Apr 4 at 10:10
• Sorry, I found that this is merely a result from the first part of the proof. – Steve Apr 4 at 10:16