In Hatcher's book, exercise: 1.1.16-c:
- Show that there are no retractions $r :X \rightarrow A$ in the following cases:
(c) $X = S_1 × D_2$ and A the circle shown in the figure.
By the inclusion, the induced map from the fundamental group of A to the fundamental group of $X$ , and the generator of the former is sent to just the homotopy class of the knotted circle inside the solid torus. But is that true that"the knot can be homotoped away by pulling the ends through each other,"? Why? Anyway who can give me a careful proof of this one? Thank you very much!