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Let $\gamma$ be a smooth, simple, closed curve and let $f : \gamma \to S^1$ assign to each $x \in \gamma$ the unit normal vector there. We can find a diffeomorphism $g: \gamma \to S^1$ and define the map $f': S^1 \to S^1$ assign to each point the unit normal. Is $f' \circ g$ (smoothly) homotopic to $f$?

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