Every surface can be triangulated in such a way that at most 7 trianlges meet at one vertex. Every surface can be decomposed in squares such that at every vertex at most 5 suqares meet. For surfaces of genus more than 1 this is the low bound.

What happen in higher dimensions, for example for 3 and 4-manifolds, ect...? It should be easy to show that for every dimension $n$ there are numbers $S(n)$ and and $C(n)$ such that every manifold $M^n$ admits a simplicial decomposition with at most $S(n)$ simplexes at every vertex and a cubical decomposition with at most $C(n)$ cubes at every vertex. The refference of Gil below confirms this for $n=3$.

Here are three questions (I suspect they are hard).

1) Can it be proven that $C(n)>2^n$?

2) Can it be proven that $S(n)>\frac{Vol(S^n)}{Vol(\Delta^n)}$, where $\Delta^n$ is the spherical tetrahedron with edge of length $\frac{\pi}{3}$ in the unit sphere $S^n$.

3) Is there any reasonable estimation for $C(n)$ and $S(n)$ from above?