I' having trouble understanding the topological genus works. It's been introduced to me via the Euler characteristic and then I've been told that, practically speaking, if the surface is orientable it counts the number of holes in it. Indeed I found another definition:
The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected (from Wikipedia).
But how is this connected to the genus definition via Euler characteristic? And is genus an actually important property? The property in itself doesn't seem so relevant, also because two surfaces with the same genus can be very different (non homeomorphic).