- Find the fundamental groups of the following spaces. In each case they can be built up from cyclic groups by free products and direct products.
(a) The space obtained from two copies of the torus $S^1\times S^1$ by identifying the simple closed curve $S^1\times\{1\}$ on the first copy of the torus with the simple closed curve $S^1 \times \{1\}$ on the second copy of the torus.
(b) The space obtained from a torus $S^1 \times S^1$ by joining two distinct points $a$ and $b$ on it by an arc which meets the torus only in its two endpoints $a$ and $b$
My trouble is with the second one, I don't know how to apply Van Kampen's theorem to this space.