Mean number of $n$-simplices per $(n-2)$-simplex in a triangulated $n$-manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T17:39:34Z http://mathoverflow.net/feeds/question/103384 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103384/mean-number-of-n-simplices-per-n-2-simplex-in-a-triangulated-n-manifold Mean number of $n$-simplices per $(n-2)$-simplex in a triangulated $n$-manifold Aaron Trout 2012-07-28T13:54:50Z 2012-07-28T14:25:10Z <p>Work by <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.ojm/1200787102" rel="nofollow">Tamura</a> (extending results by <a href="http://www.ams.org/journals/tran/1993-337-02/S0002-9947-1993-1134759-6/S0002-9947-1993-1134759-6.pdf" rel="nofollow">Luo and Stong</a>) shows the following.</p> <blockquote> <p><strong>Theorem:</strong> For any closed 3-manifold $M$ and any rational number $4.5 &lt; r &lt; 6$ there is a triangulation $T$ of $M$ for which the average edge-degree $\mu(T)$ is $r$. Here the <em>degree</em> of an edge $e$ is the number of 3-simplices having $e$ as a face.</p> </blockquote> <p>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 <strong>average bone-degree</strong></p> <p>$$\mu(T)=\frac{1}{f_{n-2}}\sum_{\tau^{n-2}\in T} \mbox{deg}(\tau^{n-2}).$$</p> <p>Here $f_k$ is the number of $k$-simplices in $T$, the sum runs over all codimension-2 simplices (called the <em>bones</em> of $T$), and deg($\tau^{n-2}$) is the number of $n$-simplices in $T$ with $\tau^{n-2}$ as a face.</p> <p>Note that by simple double-counting arguments we may alternately write this as</p> <p>$$\mu(T) = \frac{n(n+1)}{2}\frac{f_n}{f_{n-2}} = n \frac{f_{n-1}}{f_{n-2}}.$$</p> <blockquote> <p><strong>Question:</strong> 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.</p> </blockquote> <p>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. </p> <p>Thanks for any assistance pointing me in the correct direction!</p>