This is homeomorphic to a sphere no holes assuming the gluing directions are "$aa^{-1}$" for each of the edges. You can take a piece of paper and identify the edges in the given manner. You'll see that you haven't introduced an extra hole.
Otherwise, you can compute the Euler characteristic. There are 5 vertices, 4 edges, and 1 face so $\chi = 5 - 4 + 1 = 2$, and classification of surfaces tells you this is a sphere.
If you glue according to $"aa"$, then there is only 1 vertex, and the edges and faces remain the same so $\chi = 1 - 4 + 1 = -2$ so you need to check whether or not it is orientable and classification of surfaces will give you your result.