intersection in 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 $p_2in Gamma^*$. 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 $p_4in Gamma^*$. 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!
algebraic-topology manifolds dual-spaces triangulation
asked 2 days ago
wlnwln
254
add a comment |
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 $p_2in Gamma^*$. 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 $p_4in Gamma^*$. 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!
algebraic-topology manifolds dual-spaces triangulation
asked 2 days ago
wlnwln
254
add a comment |
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 $p_2in Gamma^*$. 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 $p_4in Gamma^*$. 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!
algebraic-topology manifolds dual-spaces triangulation
asked 2 days ago
wlnwln
254
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 $p_2in Gamma^*$. 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 $p_4in Gamma^*$. 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!
algebraic-topology manifolds dual-spaces triangulation
algebraic-topology manifolds dual-spaces triangulation
asked 2 days ago
wlnwln
254
asked 2 days ago
wlnwln
254
asked 2 days ago
wlnwln
254
asked 2 days ago
wlnwln
254
asked 2 days ago
wlnwln
254
254
add a comment |
add a comment |
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