What's the easiest way to compute this thing's fundamental group?
I've been playing with it for a little while, and I'm getting $\mathbb{Z}+\mathbb{Z}$. After making the ID's I think the 1-cells are homotopic to wedge sum of two circles and we have a 2-cell attached via the map $aba^{-1}b^{-1}$?
It is a torus as you can easily see from this picture:
Hence $\pi_{1}=\mathbb{Z}\oplus\mathbb{Z}$