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!

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!

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, $\mathbf{f}(T) = \left( f_0, f_1, \ldots, f_k \right)$, so perhaps someone familiar with this work can help.

Thanks for any assistance pointing me in the correct direction!

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!