Show that the fundamental group of the 2-torus $S^{1} \times S^{1}$ is isomorphic to $\mathbf{Z} \times \mathbf{Z}$.

Show that an injective continuous map from the circle $S^{1}$ to itself induces multiplication by $\pm 1$ on the fundamental group.

Show that there is no retraction from the solid torus $S^{1} \times D^{2}$ to its boundary.