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Oct 11, 2023 at 17:47 comment added Tyler Lawson Ah, silly me. Every time we have a point on $[0,1] \times \{0\}$ whose image is in $L \cap R$, transversality implies that the points on one side map to $L$ and the other map to $R$ (so it "switches sides"). Since these points start in $L$ and end in $R$, there must be an odd number of such switches, and so at least one point on the bottom has to flow to the top.
Oct 11, 2023 at 16:15 history answered Tyler Lawson CC BY-SA 4.0