# Mean number of $n$-simplices per $(n-2)$-simplex in a triangulated $n$-manifold

Work by Tamura (extending results by Luo and Stong) shows the following.

Theorem: For any closed 3-manifold $M$ and any rational number $4.5 < r < 6$ there is a triangulation $T$ of $M$ for which the average edge-degree $\mu(T)$ is $r$. Here the degree of an edge $e$ is the number of 3-simplices having $e$ as a face.

Now, suppose $T$ is a triangulation of a closed $n$-manifold $M$. (NOTE: By triangulation, I mean a combinatorial $n$-manifold, but results using a more relaxed definition would be great.) Let us extend the notion of average edge-degree to something I like to call the average bone-degree

$$\mu(T)=\frac{1}{f_{n-2}}\sum_{\tau^{n-2}\in T} \mbox{deg}(\tau^{n-2}).$$

Here $f_k$ is the number of $k$-simplices in $T$, the sum runs over all codimension-2 simplices (called the bones of $T$), and deg($\tau^{n-2}$) is the number of $n$-simplices in $T$ with $\tau^{n-2}$ as a face.

Note that by simple double-counting arguments we may alternately write this as

$$\mu(T) = \frac{n(n+1)}{2}\frac{f_n}{f_{n-2}} = n \frac{f_{n-1}}{f_{n-2}}.$$

Question: Does anyone know of results similar to the theorem above but for $n\geq 4$? That is, have the possible values for $\mu(T)$ been characterized when $n\geq 4$? Results for any manifold $M$ would be great.

I seem to remember stumbling across something like this for $n=4$ once, but I can no longer find it. I know that there is a large body of research on the "f-vectors" of manifolds, $\mathbf{f}(T) = \left( f_0, f_1, \ldots, f_n \right)$, so perhaps someone familiar with this work can help.

Thanks for any assistance pointing me in the correct direction!

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How far can you get by considering products of lower-dimensional manifolds? – Tom Goodwillie Jul 28 '12 at 18:03
@Tom Goodwillie: Is there a standard way to triangulate $M\times N$ given triangulations of $M$ and $N$? If so, is it easy to find the resulting $f$-vector in terms of the $f$-vectors for $N$ and $M$? If this is a well known procedure, my apologies for my ignorance! – Aaron Trout Jul 28 '12 at 22:13
If you use ordered triangulations, then yes there is a standard way, based on a standard way of triangulating the product of two simplices whose vertex sets are ordered. (Think of a simplex as the nerve of an ordered set so that a product of simplices appears as the nerve of the product of these ordered sets, a poset.) And it seems to me that it ought to lead to nice formulas for the $f$-vector of the product. – Tom Goodwillie Jul 29 '12 at 4:05