Example 11.2:

Let $X$ be the union of the sphere $S^2$ and one of its diameters. Then $X$ is homotopy equivalent to the space $S^2\vee S^1$. While it is easy to construct the map $f:X\longrightarrow S^2\vee S^1$, the map $g$ in the opposite direction is not easy to write down. Exercise 6 shows how to get around the difficulty.