  The Poincare-Hopf theorem tells us that given a smooth, structurally stable
vector field on a surface of genus g, the number of saddles is 2-2g less than
the number of sinks and sources. We generalize this result by introducing a
more complex combinatorial invariant. Using this tool, we demonstrate that many
such structurally stable vector fields are equivalent up to a set of basic
operations. We show in particular that for the sphere, all such vector fields
are equivalent.

Differential Geometry