The double torus:

By writing out the attaching map $S^1\longrightarrow S^1\vee S^1\vee S^1\vee S^1$ akin to (25.5) or else using the identification of the sides of a regular octagon as described in lecture 4, the reader is invited to prove that the fundamental group of the double torus is

$$\langle a, b, c, d\;\vert\; abcda^{-1}b^{-1}c^{-1}d^{-1} = 1\rangle \eqno(26.14)$$