I'm doing exercise 16 on page 39 in Hatcher:
Show that there are no retractions $r: X \rightarrow A$ in the following cases:
(a) $X = \mathbb{R}^3$ with $A$ any subspace homeomorphic to $S^1$
(b) $X = S^1 \times D^2$ with $A$ its boundary torus $S^1 \times S^1$
(c) $X = S^1 \times D^2$ and $A$ the circle shown in the figure.
I've done (a) and (b) using proposition 1.17. i.e. I assumed there was a retraction, then the map between the fundamental groups has got to be injective, therefore contradiction.
Now I'm stuck with (c) because according to my understanding the circle on the picture has the same fundamental group as $S^1$ which means it also has the same fundamental group as the solid torus.
What is a different way of proving that there is no retraction (not using prop. 1.17.)? Many thanks for your help!