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Let D be the closed unit disc in $\mathbb R^2$ and $S^1= \partial D$. Let $f$, $g$: $S^1 \rightarrow \mathbb R^3$ be smooth embeddings s.t. $f(S^1) \cap g(S^1) = \emptyset$. Define $\lambda: S^1 \times S^1 \rightarrow S^2 $ by

$$\lambda(x,y)=\frac{f(x)-g(x)}{\|f(x)-g(x)\|} $$

$1)$ Show $\lambda $ is a smooth map.

$2)$ suppose $f$ extends to a smooth map $\hat f: D \rightarrow \mathbb R^3$ with $\hat f(D) \cap g(S^1) = \emptyset$. Show that the degree of $\lambda $ is $0$.

$3)$ Let $(x_1,x_2)$ be coordinates on $S^1 \subset \mathbb R^2$ and suppose $f(x_1,x_2)=(x_1,x_2,0)$ and $g(x_1,x_2)=(0,x_1+1,x_2)$. Show that the degree of $\lambda $ is not $0$.

This is an old qual problem. Can someone give me some ideas?

WWK
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1 Answers1

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Hints:

$(1)$: As the quotient of smooth functions, it suffices to check that the denominator is never zero.

$(2)$: Show that there is a homotopy of $f$ to a constant that induces a homotopy of $\lambda$ to a map $\lambda'$ that is really only a function of a single variable, so $d\lambda'$ is never surjective. Apply Sard's theorem to compute the degree of $\lambda'$, and finally use homotopy invariance of degree to make a conclusion about the degree of $\lambda$.

$(3)$: Compute $\lambda$, and try to find the number of preimage points at a regular value.

Elchanan Solomon
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