this is for a homework problem, although not the problem itself, and I'm looking for a little guidance.
In the problem, I am given a very long list of triangles, approximately 40, and asked to classify the surface using the orientability of the surface as well as it's Euler characteristic. It doesn't seem to me hard to find the characteristic - the faces, edges and vertices are given clearly in the triangulation, even if it is a bit of a pain to figure out exactly how many edges there are. However, once I have this number, I have to decide between one of two possibilities, depending on whether or not the surface is orientable, and if the surface has boundary, I have to determine how many disks, if any, have been cut out.
Is there a way to do this that does not involve trying to draw the complex with all 40 some odd triangles, and rearranging them endlessly? In simpler examples, this, in conjunction with cut/paste arguments has been efficient, but I don't see a reasonable way to go about this problem with so many triangles.
One thing you have to check ahead of time is connectivity; I'll presume that has been done.
To determine orientability, do a depth first search to construct a polygon out of the 40 triangles. It will take 39 gluings. The resulting polygon will have 42 sides, glued in pairs. Check whether any of the 21 edge pair gluings reverse orientation.
Alternatively, construct the chain complex $C_2$ with a basis of 40 2-simplices, and the chain complex $C_1$ with a basis of 60 1-simplices, and construct the boundary map, an extremely sparse 40-by-60 matrix, and compute its kernel, the group of cycles. The kernel will be either $\mathbb{Z}$ and it's orientable, or $0$ and it's non-orientable.
(These are really the same method, stated in different ways)