I'm studying some oriented geometries in the Grassmannian. These geometries, even if their interior is connected and oriented, have some funny behaviour on their boundaries. For example, some one-dimensional boundaries look like two segments with non-matching orientations, that is
So, my question is: Does it exist in the literature a natural definition of the Euler characteristic for this type of object?
I would say that the union of these two segments has 2 interiors (or faces), 2 external boundaries (points 1 and 3) and 1 internal boundary, that is point 2, where the two orientations don't match and that in my opinion, it makes sense to count twice. So the Euler characteristic can be computed as χ = 2 - 4 =2, which is the same as the Euler characteristic of 2 segments.
I found in this paper that they define something called orientation number and use it to define the Euler characteristic as a self-intersection number. It looks relevant, but I'm a physicist and I'm having a hard time understanding it.
Any help will be appreciated.