In dimension 2, the euler poincare formula restricts the incidence properties of edges in a triangulation of a surface. 
Are there analogous generalizations for higher dimensions, like elaborations on the Schenzel formula for the specific case pf spheres?



f vectors of simplicial complexes homeomorphic to n dimensional spheres
The $g$-conjecture (or $g$-theorem in the polytopal case) aims the characterize what possible $f$-vectors can arise from simplicial spheres. [Here](https://gilkalai.wordpress.com/2009/04/04/how-the-g-conjecture-came-about/) is some nice history and introduction to the $g$-conjecture by Gil Kalai. In brief the $g$-conjecture says that for a simplicial sphere the $h$-vector should be symmetric and the $g$-vector should be an $M$-vector where $g\_i = h\_i - h\_{i-1}$.


