The following is an exercise in computing homology:
Let $K$ be the union of the boundaries of two 2-simplices, joined along one edge. Compute the homology of $K$.
Since the standard 2-simplex is a triangle in $\mathbb{R}^3$, I was thinking pictorially, this gluing would result in either a diamond or two triangles joined by a line segment whose endpoints are vertices, one from each triangle. My knee-jerk guess is that the first one is the right picture, but I am hoping someone could set me straight here.
Beyond thinking about a picture, I am not sure where to go on this problem. If anyone visiting the site today is up for giving me a jump start/ walking me through an approach to this problem, that would be greatly appreciated.