Generalising dual triangulation of manifolds
$begingroup$
We know the following “geometric” version of Poincaré duality:
Let $M$ be a closed $m$-dimensional manifold and let $mathfrak{X}_*$ be a finite simplicial complex with $|mathfrak{X}_*|=M$. We can build a dual cell complex $mathfrak{X}^*$: For each $p$-cell $Sigma$ we consider as the dual cell $D(Sigma)$ the convex hull of the barycenters of all $m$-simplices $Sigma'$ with $SigmapreceqSigma'$. Then $|mathfrak{X}^*|cong M$.
The cells in the dual decomposition are not necessarily simplices, but polyhedra. The master example is the triangulation of $mathbb{S}^2$ as the surface of the octahedron. The dual decomposition is a cube.
I am looking for a good reference for this. Even more, I want to generalise the statement to other sorts of “simplicial complexes”, where we are allowed to contract faces. (Take e. g. the Fuks complex for the one-point compactification of the unordered configuration space $C^k(mathbb{C})$, The cells are products of simplices and many boundaries are degenerate and hence contracted to the $infty$-point.) I would like to have the following statement:
Let $M$ be an open $m$-dimensional manifold and $mathfrak{X}_*$ a finite “triangulation” of the one-point compactification $Mcup{infty}$ where some faces may be contracted to the $0$-cell $infty$. Then there is a dual cell complex $mathfrak{X}^*$ with $mathfrak{X}^pcong mathfrak{X}_{m-p}$ and $$mathfrak{X}^pcongbegin{cases}mathfrak{X}_{m-p} &text{for } 0le ple m-1,\mathfrak{X}_0setminus{infty} &text{for } p=m.end{cases}$$
This dual complex has the geometric realisation homotopy equivalent to $M$: $|mathfrak{X}^*|simeq M$
Something like this must exist, since it is used often as an argument for the homotopy type of $C^k(mathbb{C})$: The Fuks complex has only cells in dimension $k+1le dle 2k$ and the $0$-cell $infty$. Thus, the dual is a cellular complex of dimension $k-1$. Furthermore, the usage of the one-point compactification reminds me at the Poincaré–Lefschetz duality $H^p(M)cong H_{m-p}(Mcup{infty},infty)$ for orientable $M$.
I would also be happy with some hints to literature about this “old”, geometric approach to duality.
manifolds simplicial-complex poincare-duality configuration-space
asked Dec 6 '18 at 18:05
FKranholdFKranhold
1987
$endgroup$
add a comment |
$begingroup$
We know the following “geometric” version of Poincaré duality:
Let $M$ be a closed $m$-dimensional manifold and let $mathfrak{X}_*$ be a finite simplicial complex with $|mathfrak{X}_*|=M$. We can build a dual cell complex $mathfrak{X}^*$: For each $p$-cell $Sigma$ we consider as the dual cell $D(Sigma)$ the convex hull of the barycenters of all $m$-simplices $Sigma'$ with $SigmapreceqSigma'$. Then $|mathfrak{X}^*|cong M$.
The cells in the dual decomposition are not necessarily simplices, but polyhedra. The master example is the triangulation of $mathbb{S}^2$ as the surface of the octahedron. The dual decomposition is a cube.
I am looking for a good reference for this. Even more, I want to generalise the statement to other sorts of “simplicial complexes”, where we are allowed to contract faces. (Take e. g. the Fuks complex for the one-point compactification of the unordered configuration space $C^k(mathbb{C})$, The cells are products of simplices and many boundaries are degenerate and hence contracted to the $infty$-point.) I would like to have the following statement:
Let $M$ be an open $m$-dimensional manifold and $mathfrak{X}_*$ a finite “triangulation” of the one-point compactification $Mcup{infty}$ where some faces may be contracted to the $0$-cell $infty$. Then there is a dual cell complex $mathfrak{X}^*$ with $mathfrak{X}^pcong mathfrak{X}_{m-p}$ and $$mathfrak{X}^pcongbegin{cases}mathfrak{X}_{m-p} &text{for } 0le ple m-1,\mathfrak{X}_0setminus{infty} &text{for } p=m.end{cases}$$
This dual complex has the geometric realisation homotopy equivalent to $M$: $|mathfrak{X}^*|simeq M$
Something like this must exist, since it is used often as an argument for the homotopy type of $C^k(mathbb{C})$: The Fuks complex has only cells in dimension $k+1le dle 2k$ and the $0$-cell $infty$. Thus, the dual is a cellular complex of dimension $k-1$. Furthermore, the usage of the one-point compactification reminds me at the Poincaré–Lefschetz duality $H^p(M)cong H_{m-p}(Mcup{infty},infty)$ for orientable $M$.
I would also be happy with some hints to literature about this “old”, geometric approach to duality.
manifolds simplicial-complex poincare-duality configuration-space
asked Dec 6 '18 at 18:05
FKranholdFKranhold
1987
$endgroup$
add a comment |
$begingroup$
We know the following “geometric” version of Poincaré duality:
Let $M$ be a closed $m$-dimensional manifold and let $mathfrak{X}_*$ be a finite simplicial complex with $|mathfrak{X}_*|=M$. We can build a dual cell complex $mathfrak{X}^*$: For each $p$-cell $Sigma$ we consider as the dual cell $D(Sigma)$ the convex hull of the barycenters of all $m$-simplices $Sigma'$ with $SigmapreceqSigma'$. Then $|mathfrak{X}^*|cong M$.
The cells in the dual decomposition are not necessarily simplices, but polyhedra. The master example is the triangulation of $mathbb{S}^2$ as the surface of the octahedron. The dual decomposition is a cube.
I am looking for a good reference for this. Even more, I want to generalise the statement to other sorts of “simplicial complexes”, where we are allowed to contract faces. (Take e. g. the Fuks complex for the one-point compactification of the unordered configuration space $C^k(mathbb{C})$, The cells are products of simplices and many boundaries are degenerate and hence contracted to the $infty$-point.) I would like to have the following statement:
Let $M$ be an open $m$-dimensional manifold and $mathfrak{X}_*$ a finite “triangulation” of the one-point compactification $Mcup{infty}$ where some faces may be contracted to the $0$-cell $infty$. Then there is a dual cell complex $mathfrak{X}^*$ with $mathfrak{X}^pcong mathfrak{X}_{m-p}$ and $$mathfrak{X}^pcongbegin{cases}mathfrak{X}_{m-p} &text{for } 0le ple m-1,\mathfrak{X}_0setminus{infty} &text{for } p=m.end{cases}$$
This dual complex has the geometric realisation homotopy equivalent to $M$: $|mathfrak{X}^*|simeq M$
Something like this must exist, since it is used often as an argument for the homotopy type of $C^k(mathbb{C})$: The Fuks complex has only cells in dimension $k+1le dle 2k$ and the $0$-cell $infty$. Thus, the dual is a cellular complex of dimension $k-1$. Furthermore, the usage of the one-point compactification reminds me at the Poincaré–Lefschetz duality $H^p(M)cong H_{m-p}(Mcup{infty},infty)$ for orientable $M$.
I would also be happy with some hints to literature about this “old”, geometric approach to duality.
manifolds simplicial-complex poincare-duality configuration-space
asked Dec 6 '18 at 18:05
FKranholdFKranhold
1987
$endgroup$
We know the following “geometric” version of Poincaré duality:
Let $M$ be a closed $m$-dimensional manifold and let $mathfrak{X}_*$ be a finite simplicial complex with $|mathfrak{X}_*|=M$. We can build a dual cell complex $mathfrak{X}^*$: For each $p$-cell $Sigma$ we consider as the dual cell $D(Sigma)$ the convex hull of the barycenters of all $m$-simplices $Sigma'$ with $SigmapreceqSigma'$. Then $|mathfrak{X}^*|cong M$.
The cells in the dual decomposition are not necessarily simplices, but polyhedra. The master example is the triangulation of $mathbb{S}^2$ as the surface of the octahedron. The dual decomposition is a cube.
I am looking for a good reference for this. Even more, I want to generalise the statement to other sorts of “simplicial complexes”, where we are allowed to contract faces. (Take e. g. the Fuks complex for the one-point compactification of the unordered configuration space $C^k(mathbb{C})$, The cells are products of simplices and many boundaries are degenerate and hence contracted to the $infty$-point.) I would like to have the following statement:
Let $M$ be an open $m$-dimensional manifold and $mathfrak{X}_*$ a finite “triangulation” of the one-point compactification $Mcup{infty}$ where some faces may be contracted to the $0$-cell $infty$. Then there is a dual cell complex $mathfrak{X}^*$ with $mathfrak{X}^pcong mathfrak{X}_{m-p}$ and $$mathfrak{X}^pcongbegin{cases}mathfrak{X}_{m-p} &text{for } 0le ple m-1,\mathfrak{X}_0setminus{infty} &text{for } p=m.end{cases}$$
This dual complex has the geometric realisation homotopy equivalent to $M$: $|mathfrak{X}^*|simeq M$
Something like this must exist, since it is used often as an argument for the homotopy type of $C^k(mathbb{C})$: The Fuks complex has only cells in dimension $k+1le dle 2k$ and the $0$-cell $infty$. Thus, the dual is a cellular complex of dimension $k-1$. Furthermore, the usage of the one-point compactification reminds me at the Poincaré–Lefschetz duality $H^p(M)cong H_{m-p}(Mcup{infty},infty)$ for orientable $M$.
I would also be happy with some hints to literature about this “old”, geometric approach to duality.
manifolds simplicial-complex poincare-duality configuration-space
manifolds simplicial-complex poincare-duality configuration-space
asked Dec 6 '18 at 18:05
FKranholdFKranhold
1987
asked Dec 6 '18 at 18:05
FKranholdFKranhold
1987
asked Dec 6 '18 at 18:05
FKranholdFKranhold
1987
asked Dec 6 '18 at 18:05
FKranholdFKranhold
1987
asked Dec 6 '18 at 18:05
FKranholdFKranhold
1987
1987
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