Show that any triangulation of a compact Riemann surface is orientable.
Hint. Since one can take suitable refinements, there is no loss of generality
in assuming that the union of two triangles with a joint edge is contained in the
domain of definition of an analytic chart. Define the orientation of a triangle in
such a way that the winding numbers with respect to the analytic chart around
inner points are +1. This possible because of the result of Exercise 3. It follows
from the result of Exercise 2 that this orientation is independent of the choice of
the analytic chart. Applying the result of Exercise 2 once more, we can see that
an orientation of the triangulation is obtained.