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?
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$\begingroup$ For a triangulated manifold of dimension $d > 0$, the $(d-1)$-dimensional simplices must all have exactly two $d$-dimensional co-faces. $\endgroup$– Vidit NandaCommented Dec 7, 2015 at 17:42
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2 Answers
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The $g$-conjecture (or $g$-theorem in the polytopal case) aims the characterize what possible $f$-vectors can arise from simplicial spheres. Here 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 $g_i = h_i - h_{i-1}$.
answered Dec 7, 2015 at 23:23
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As is alluded to in the article linked in the previous answer, the Dehn-Sommerville equations are a powerful generalization of the Euler-Poincare formula for a homology sphere. These say that, for a homology $d$-sphere, the relation $h_i = h_{d-i}$ holds for every $i \in {0,\dots, d}$. This simple $h$-vector relation can be expanded out into a rather more complicated (but equivalent) relation on the $f$-vector. In particular, the relation $h_d = h_0$ recovers the Euler-Poincare formula.
There's a nice Wikipedia article on the Dehn-Sommerville equations. In particular, you can look here if you want to see the $f$-vector formulation of the equations.
The Wikipedia article only mentions simplicial polytopes. If you want a reference for the statement on homology spheres, see the Kalai article (informally), or, in somewhat different language, Stanley's book Combinatorics and Commutative Algebra.
answered Dec 8, 2015 at 1:30