# dual and intersection of a simplex

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!