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It is a well known result in algebraic topology that there is no retraction of $D^2$ onto $S^1$.

Does anyone know any continuous maps $D^2 \to S^1$ which are not constant?

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    @Dbr: Your map would be a map $S^1\to S^1$ which factors through $D^2$. Since $D^2$ is contractible, your original map is null, hence degree 0.2011-05-14

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How about the map that sends $(x,y)\in D^2=\{(x,y)\in\mathbb{R}^2\mid x^2+y^2\leq 1\}$ to $(\textstyle\sqrt{1-y^2},y)\in S^1=\{(x,y)\in\mathbb{R}^2\mid x^2+y^2=1\}$

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If you can think of a path $\gamma\colon I \to S^1$, then you get a continuous map $D^2 \to S^1$ by first collapsing $D^2$ to the interval $I$ and then composing with your path.

If your path $\gamma$ was not constant, then the resulting map $D^2 \to S^1$ will not be constant.