Vector field on n-manifold whose sum of indexes is equal to Euler charasteristic

Question

For 2-manifolds and 3-manifolds such a tangent field (whose singular points indexes sum to manifold's Euler chracteristic) construction can be done visually. For example, for triangulated 2-manifold it's shown in the following picture

Is there a universal formula for the field in higher dimentions?

Answer 1

There is a such formula for manifolds. A good refernece is Thurston's Three-Dimensional Geometry and Topology, which he did this explicitly by coloring the vertices for the surfaces. There is no spoiler here, as Thurston only gave hints, not complete solution to the problem. Enjoy!