We know that exists a non vanishing vector field on the torus, if we want as a consequence of the Poincarè-Hopf Theorem.
But what if we remove a disk from the torus ? Is it combable ? We define a vector field on this manifold with boundary to be "tangent", if the vector field points in the outward direction with respect to the boundary.
I was thinking, if it has to go "outward" to the boundary it means that "points" in the direction on a zero that should look like a zero of non $0$ index (maybe this intuition is false), so if I think to this vector adding back the disk I should have another $0$ to cancel out this one, because we know that the characteristic of Euler of the torus in $0$. Is this true ?
My guess is that this manifold is not combable but I don't how to prove. And I also think that solved this case maybe this could be generalized to any genus $g$ torus.
Any help would be appreciated