# Computing homology of the boundary of two “bonded” 2-simplices

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.

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Another way I would describe it: a square frame with a segment connecting one diagonal. I think this is homotopy equivalent to $S^1 \vee S^1$. – Dylan Moreland Feb 19 '12 at 21:34
Here are some suggestions on getting started: 1) Label all of your 0-simplices and 1-simplices. 2) Choose orientations for each 1-simplex. 3) Write down the simplicial chain complex for $K$. The orientations you chose will determine the boundary map. 4) Compute homology. – wckronholm Feb 19 '12 at 22:14