Let $f : D \rightarrow D$ be a continuous map whose restriction to $S^1$ is the identity map. Show that $f$ must be surjective.

Question

Let $D$ denote the closed unit disc in the plane with boundary the unit circle $S^1$. Let $f : D \rightarrow D$ be a continuous map whose restriction to $S^1$ is the identity map. Show that $f$ must be surjective.

Answer 1

If the map omitted the point $p$ in the disk, you could compose the given map $f$ with a continuous projection from $p$ to the boundary of the disk ($S^1$). This composition gives a retraction $D\rightarrow S^1$. It is a standard exercise that such a retraction cannot exist. Use your favorite invariant -- homology, the fundamental group, whatever -- to show this. The key fact is that the retraction is going to induce a surjection on your chosen invariant.