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In a triangulation $\Gamma$ of a (oriented) 2-manifold, consider a 2-simplex labeled by ($123$), where $1,2,3$ denote the order of vertices. Consider the dual $\Gamma^*$ of $\Gamma$, and then denote the dual of the 2-simplex $(123)$ as $p_2\in \Gamma^*$. Further consider the dual of the 1-simplex $(12)$, $(23)$, $(13)$ $\in$ $\Gamma$, which are denoted as $(\overline{12}), (\overline{23}), (\overline{13})\in \Gamma^*$. It can be easily see that $(\overline{12}) \cap (\overline{23})=(\overline{12})\cap (\overline{13})=(\overline{13})\cap (\overline{13})=p_2$.

My questions are about the generalization of the above observation.

(1) The first question is about the generalization to $2n$-manifold. Also denote a triangulation of the (oriented)$2n$-manifold as $\Gamma$ whose dual as $\Gamma^*$. Consider a $2n$-simplex $(i_1i_2...i_{2n+1})\in \Gamma$ whose dual is $p_{2n}\in \Gamma^*$. Further denote the dual of two n-simplices $(i_1i_2...i_{n+1}), (i_{n+1}i_{n+2}...i_{2n+1})\in \Gamma$ as $(\overline{i_1i_2...i_{n+1}}), (\overline{i_{n+1}i_{n+2}...i_{2n+1}})\in \Gamma^*$, respectively. The question is that whether $(\overline{i_1i_2...i_{n+1}})\cap (\overline{i_{n+1}i_{n+2}...i_{2n=1}})=p_{2n+1}$. For example the 4-simplex case. Consider a 4-simplex $(12345)\in \Gamma$ whose dual is $p_4\in \Gamma^*$. The dual of the 2-simplex $(123), (345)\in \Gamma$ is $(\overline{123}), (\overline{345})\in \Gamma^*$. The question is that whether $(\overline{123})\cap (\overline{345})=p_4$.

(2) The second question is about the generalization to $2n-1$-manifold. Now the simplex is denoted as $(i_1i_2...i_{2n}) \in \Gamma$ whose dual is $p_{2n-1}\in \Gamma^*$. The dual of two subsimplices $(i_1i_2...i_{n+1}), (i_{n+1}i_{n+2}...i_{2n})\in \Gamma$ as $(\overline{i_1i_2...i_{n+1}}), (\overline{i_{n+1}i_{n+2}...i_{2n}})\in \Gamma^*$. Similarly, whether $(\overline{i_1i_2...i_{n+1}}) \cap (\overline{i_{n+1}i_{n+2}...i_{2n}})=p_{2n-1}\in \Gamma^*$?

Observe that $(12)$ and $(23)$ intersect "transversely" at a point(similar to the other choices) in the 2-manifold and also $(123), (345)$ intersect "transversely" at a point in the 4-manifold. Now we boldly conjecture that

In a (oriented) n-manifold, the dual of two subsimplex ($i_1i_2...i_k$) and $(i_{k}i_{k+1}...i_{n+1})$ of a n-simplex ($i_1i_2...i_{n+1}$) intersect at the dual of the n-simplex

Is this conjecture correct generally or under what condition(s)?

Any reference or suggestion or idea is welcome. Thanks!

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