cut and glue technique for higher dimensional complexes
$begingroup$
My understanding is that with surfaces, it is possible to start from a 2D sheet and use cut and glue to construct all possible surfaces up to Homeomorphism. That is cut and glue is as expressive as 2D simplicial complexes in its expressive power.
What type of higher dimensional complexes can we express using cut and glue in higher dimensions?
Are there any good references for this?
general-topology algebraic-topology
asked Dec 20 '18 at 0:32
self-educatorself-educator
3911
$endgroup$
|
show 2 more comments
$begingroup$
My understanding is that with surfaces, it is possible to start from a 2D sheet and use cut and glue to construct all possible surfaces up to Homeomorphism. That is cut and glue is as expressive as 2D simplicial complexes in its expressive power.
What type of higher dimensional complexes can we express using cut and glue in higher dimensions?
Are there any good references for this?
general-topology algebraic-topology
asked Dec 20 '18 at 0:32
self-educatorself-educator
3911
$endgroup$
1
$begingroup$
Surfaces are to a ridiculous but fantastic degree much simpler than higher dimensional manifolds. There is no hope to say anything reasonable about higher dimensional manifolds by the same techniques.
$endgroup$
– Mike Miller
Dec 20 '18 at 2:15
1
$begingroup$
There should be a class of complexes that we can build using this approach (?) For example, it seems to me that N-dimensional Torus and Sphere can be built this way.
$endgroup$
– self-educator
Dec 20 '18 at 3:21
1
$begingroup$
Your question as asked in un-answerable since you did not specify what cut and glue operations your are allowing. I suggest you first work out what your question is for discrete topological spaces (0-dimensional simplicial complexes), then for simplicial graphs. Also, keep in mind that for each connected n-dimensional triangulated manifold $M$, you can obtain $M$ by gluing some boundary faces of a triangulated $n$-dimensional ball. Maybe this is what you wanted to know.
$endgroup$
– Moishe Cohen
Dec 25 '18 at 22:51
$begingroup$
I'm interested in building n-dimensional simplicial complexes using n-dimensional polytopes + cut and glue (analogous to the role of cut and glue in building surfaces). In particular, I'm wondering what type of n-dimensional complexes we can/can't build, starting from an arbitrary n-dimensional polytope and gluing faces together.
$endgroup$
– self-educator
Dec 26 '18 at 5:36
1
$begingroup$
Still unclear. But it seems that you are allowing just one polytope for each space you are building. Then, as I said, you can get any compact connected triangulated manifold (possibly after subdividing the triangulation since you are insisting on polytopes). You can get many other complexes (already in dimension 2). There is no good description besides tautological one.
$endgroup$
– Moishe Cohen
Dec 27 '18 at 3:19
|
show 2 more comments
$begingroup$
My understanding is that with surfaces, it is possible to start from a 2D sheet and use cut and glue to construct all possible surfaces up to Homeomorphism. That is cut and glue is as expressive as 2D simplicial complexes in its expressive power.
What type of higher dimensional complexes can we express using cut and glue in higher dimensions?
Are there any good references for this?
general-topology algebraic-topology
asked Dec 20 '18 at 0:32
self-educatorself-educator
3911
$endgroup$
My understanding is that with surfaces, it is possible to start from a 2D sheet and use cut and glue to construct all possible surfaces up to Homeomorphism. That is cut and glue is as expressive as 2D simplicial complexes in its expressive power.
What type of higher dimensional complexes can we express using cut and glue in higher dimensions?
Are there any good references for this?
general-topology algebraic-topology
general-topology algebraic-topology
asked Dec 20 '18 at 0:32
self-educatorself-educator
3911
asked Dec 20 '18 at 0:32
self-educatorself-educator
3911
asked Dec 20 '18 at 0:32
self-educatorself-educator
3911
asked Dec 20 '18 at 0:32
self-educatorself-educator
3911
asked Dec 20 '18 at 0:32
self-educatorself-educator
3911
3911
1
$begingroup$
Surfaces are to a ridiculous but fantastic degree much simpler than higher dimensional manifolds. There is no hope to say anything reasonable about higher dimensional manifolds by the same techniques.
$endgroup$
– Mike Miller
Dec 20 '18 at 2:15
1
$begingroup$
There should be a class of complexes that we can build using this approach (?) For example, it seems to me that N-dimensional Torus and Sphere can be built this way.
$endgroup$
– self-educator
Dec 20 '18 at 3:21
1
$begingroup$
Your question as asked in un-answerable since you did not specify what cut and glue operations your are allowing. I suggest you first work out what your question is for discrete topological spaces (0-dimensional simplicial complexes), then for simplicial graphs. Also, keep in mind that for each connected n-dimensional triangulated manifold $M$, you can obtain $M$ by gluing some boundary faces of a triangulated $n$-dimensional ball. Maybe this is what you wanted to know.
$endgroup$
– Moishe Cohen
Dec 25 '18 at 22:51
$begingroup$
I'm interested in building n-dimensional simplicial complexes using n-dimensional polytopes + cut and glue (analogous to the role of cut and glue in building surfaces). In particular, I'm wondering what type of n-dimensional complexes we can/can't build, starting from an arbitrary n-dimensional polytope and gluing faces together.
$endgroup$
– self-educator
Dec 26 '18 at 5:36
1
$begingroup$
Still unclear. But it seems that you are allowing just one polytope for each space you are building. Then, as I said, you can get any compact connected triangulated manifold (possibly after subdividing the triangulation since you are insisting on polytopes). You can get many other complexes (already in dimension 2). There is no good description besides tautological one.
$endgroup$
– Moishe Cohen
Dec 27 '18 at 3:19
|
show 2 more comments
1
$begingroup$
Surfaces are to a ridiculous but fantastic degree much simpler than higher dimensional manifolds. There is no hope to say anything reasonable about higher dimensional manifolds by the same techniques.
$endgroup$
– Mike Miller
Dec 20 '18 at 2:15
1
$begingroup$
There should be a class of complexes that we can build using this approach (?) For example, it seems to me that N-dimensional Torus and Sphere can be built this way.
$endgroup$
– self-educator
Dec 20 '18 at 3:21
1
$begingroup$
Your question as asked in un-answerable since you did not specify what cut and glue operations your are allowing. I suggest you first work out what your question is for discrete topological spaces (0-dimensional simplicial complexes), then for simplicial graphs. Also, keep in mind that for each connected n-dimensional triangulated manifold $M$, you can obtain $M$ by gluing some boundary faces of a triangulated $n$-dimensional ball. Maybe this is what you wanted to know.
$endgroup$
– Moishe Cohen
Dec 25 '18 at 22:51
$begingroup$
I'm interested in building n-dimensional simplicial complexes using n-dimensional polytopes + cut and glue (analogous to the role of cut and glue in building surfaces). In particular, I'm wondering what type of n-dimensional complexes we can/can't build, starting from an arbitrary n-dimensional polytope and gluing faces together.
$endgroup$
– self-educator
Dec 26 '18 at 5:36
1
$begingroup$
Still unclear. But it seems that you are allowing just one polytope for each space you are building. Then, as I said, you can get any compact connected triangulated manifold (possibly after subdividing the triangulation since you are insisting on polytopes). You can get many other complexes (already in dimension 2). There is no good description besides tautological one.
$endgroup$
– Moishe Cohen
Dec 27 '18 at 3:19
1
1
$begingroup$
Surfaces are to a ridiculous but fantastic degree much simpler than higher dimensional manifolds. There is no hope to say anything reasonable about higher dimensional manifolds by the same techniques.
$endgroup$
– Mike Miller
Dec 20 '18 at 2:15
$begingroup$
Surfaces are to a ridiculous but fantastic degree much simpler than higher dimensional manifolds. There is no hope to say anything reasonable about higher dimensional manifolds by the same techniques.
$endgroup$
– Mike Miller
Dec 20 '18 at 2:15
1
1
$begingroup$
There should be a class of complexes that we can build using this approach (?) For example, it seems to me that N-dimensional Torus and Sphere can be built this way.
$endgroup$
– self-educator
Dec 20 '18 at 3:21
$begingroup$
There should be a class of complexes that we can build using this approach (?) For example, it seems to me that N-dimensional Torus and Sphere can be built this way.
$endgroup$
– self-educator
Dec 20 '18 at 3:21
1
1
$begingroup$
Your question as asked in un-answerable since you did not specify what cut and glue operations your are allowing. I suggest you first work out what your question is for discrete topological spaces (0-dimensional simplicial complexes), then for simplicial graphs. Also, keep in mind that for each connected n-dimensional triangulated manifold $M$, you can obtain $M$ by gluing some boundary faces of a triangulated $n$-dimensional ball. Maybe this is what you wanted to know.
$endgroup$
– Moishe Cohen
Dec 25 '18 at 22:51
$begingroup$
Your question as asked in un-answerable since you did not specify what cut and glue operations your are allowing. I suggest you first work out what your question is for discrete topological spaces (0-dimensional simplicial complexes), then for simplicial graphs. Also, keep in mind that for each connected n-dimensional triangulated manifold $M$, you can obtain $M$ by gluing some boundary faces of a triangulated $n$-dimensional ball. Maybe this is what you wanted to know.
$endgroup$
– Moishe Cohen
Dec 25 '18 at 22:51
$begingroup$
I'm interested in building n-dimensional simplicial complexes using n-dimensional polytopes + cut and glue (analogous to the role of cut and glue in building surfaces). In particular, I'm wondering what type of n-dimensional complexes we can/can't build, starting from an arbitrary n-dimensional polytope and gluing faces together.
$endgroup$
– self-educator
Dec 26 '18 at 5:36
$begingroup$
I'm interested in building n-dimensional simplicial complexes using n-dimensional polytopes + cut and glue (analogous to the role of cut and glue in building surfaces). In particular, I'm wondering what type of n-dimensional complexes we can/can't build, starting from an arbitrary n-dimensional polytope and gluing faces together.
$endgroup$
– self-educator
Dec 26 '18 at 5:36
1
1
$begingroup$
Still unclear. But it seems that you are allowing just one polytope for each space you are building. Then, as I said, you can get any compact connected triangulated manifold (possibly after subdividing the triangulation since you are insisting on polytopes). You can get many other complexes (already in dimension 2). There is no good description besides tautological one.
$endgroup$
– Moishe Cohen
Dec 27 '18 at 3:19
$begingroup$
Still unclear. But it seems that you are allowing just one polytope for each space you are building. Then, as I said, you can get any compact connected triangulated manifold (possibly after subdividing the triangulation since you are insisting on polytopes). You can get many other complexes (already in dimension 2). There is no good description besides tautological one.
$endgroup$
– Moishe Cohen
Dec 27 '18 at 3:19
|
show 2 more comments
1 Answer
1
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oldest
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$begingroup$
This construction I am sure is written somewhere, most likely, in the book H.Seifert, W.Threlfall, "A Textbook of Topology" (I just do not have it with me now):
Let $M$ be a compact connected triangulated $n$-dimensional manifold (for simplicity, without boundary). Let $G$ be the graph dual to the triangulation $Delta$: The vertices of $G$ are the barycenters of the facets (n-dimensional simplices), two vertices are connected by an edge $e$ iff the corresponding facets have a common codimension 1 face denoted $e^*$ (a "panel"). Let $Tsubset G$ be a maximal subtree. Next, "cut open" $M$ along the panels $e^*$ for the edges $e$ of $G$ which are not in $T$. The result is a finite simplicial complex $Delta_T$. One then verifies that $T$ is isomorphic to a triangulated $n$-dimensional ball. With more work one proves that some subdivision of $Delta_T$ is isomorphic to a convex simplicial (i.e. its faces define a triangulated sphere) polyhedron in $R^n$. My suggestion is to work this out in the case of surfaces, say, of a triangulated 2-dimensional sphere.
Now, one can reverse this process and glue appropriate faces of $Delta_T$ (the ones which have the same image in $M$). The result is your triangulated manifold $M$.
Edit. I finally checked: Seifert and Threlfall have the above construction in section 60 of their book, but only for 3-dimensional manifolds. The same works in all dimensions, just things become messier.
answered Dec 28 '18 at 1:53
Moishe CohenMoishe Cohen
46.1k342104
$endgroup$
add a comment |
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$begingroup$
This construction I am sure is written somewhere, most likely, in the book H.Seifert, W.Threlfall, "A Textbook of Topology" (I just do not have it with me now):
Let $M$ be a compact connected triangulated $n$-dimensional manifold (for simplicity, without boundary). Let $G$ be the graph dual to the triangulation $Delta$: The vertices of $G$ are the barycenters of the facets (n-dimensional simplices), two vertices are connected by an edge $e$ iff the corresponding facets have a common codimension 1 face denoted $e^*$ (a "panel"). Let $Tsubset G$ be a maximal subtree. Next, "cut open" $M$ along the panels $e^*$ for the edges $e$ of $G$ which are not in $T$. The result is a finite simplicial complex $Delta_T$. One then verifies that $T$ is isomorphic to a triangulated $n$-dimensional ball. With more work one proves that some subdivision of $Delta_T$ is isomorphic to a convex simplicial (i.e. its faces define a triangulated sphere) polyhedron in $R^n$. My suggestion is to work this out in the case of surfaces, say, of a triangulated 2-dimensional sphere.
Now, one can reverse this process and glue appropriate faces of $Delta_T$ (the ones which have the same image in $M$). The result is your triangulated manifold $M$.
Edit. I finally checked: Seifert and Threlfall have the above construction in section 60 of their book, but only for 3-dimensional manifolds. The same works in all dimensions, just things become messier.
answered Dec 28 '18 at 1:53
Moishe CohenMoishe Cohen
46.1k342104
$endgroup$
add a comment |
$begingroup$
This construction I am sure is written somewhere, most likely, in the book H.Seifert, W.Threlfall, "A Textbook of Topology" (I just do not have it with me now):
Let $M$ be a compact connected triangulated $n$-dimensional manifold (for simplicity, without boundary). Let $G$ be the graph dual to the triangulation $Delta$: The vertices of $G$ are the barycenters of the facets (n-dimensional simplices), two vertices are connected by an edge $e$ iff the corresponding facets have a common codimension 1 face denoted $e^*$ (a "panel"). Let $Tsubset G$ be a maximal subtree. Next, "cut open" $M$ along the panels $e^*$ for the edges $e$ of $G$ which are not in $T$. The result is a finite simplicial complex $Delta_T$. One then verifies that $T$ is isomorphic to a triangulated $n$-dimensional ball. With more work one proves that some subdivision of $Delta_T$ is isomorphic to a convex simplicial (i.e. its faces define a triangulated sphere) polyhedron in $R^n$. My suggestion is to work this out in the case of surfaces, say, of a triangulated 2-dimensional sphere.
Now, one can reverse this process and glue appropriate faces of $Delta_T$ (the ones which have the same image in $M$). The result is your triangulated manifold $M$.
Edit. I finally checked: Seifert and Threlfall have the above construction in section 60 of their book, but only for 3-dimensional manifolds. The same works in all dimensions, just things become messier.
answered Dec 28 '18 at 1:53
Moishe CohenMoishe Cohen
46.1k342104
$endgroup$
add a comment |
$begingroup$
This construction I am sure is written somewhere, most likely, in the book H.Seifert, W.Threlfall, "A Textbook of Topology" (I just do not have it with me now):
Let $M$ be a compact connected triangulated $n$-dimensional manifold (for simplicity, without boundary). Let $G$ be the graph dual to the triangulation $Delta$: The vertices of $G$ are the barycenters of the facets (n-dimensional simplices), two vertices are connected by an edge $e$ iff the corresponding facets have a common codimension 1 face denoted $e^*$ (a "panel"). Let $Tsubset G$ be a maximal subtree. Next, "cut open" $M$ along the panels $e^*$ for the edges $e$ of $G$ which are not in $T$. The result is a finite simplicial complex $Delta_T$. One then verifies that $T$ is isomorphic to a triangulated $n$-dimensional ball. With more work one proves that some subdivision of $Delta_T$ is isomorphic to a convex simplicial (i.e. its faces define a triangulated sphere) polyhedron in $R^n$. My suggestion is to work this out in the case of surfaces, say, of a triangulated 2-dimensional sphere.
Now, one can reverse this process and glue appropriate faces of $Delta_T$ (the ones which have the same image in $M$). The result is your triangulated manifold $M$.
Edit. I finally checked: Seifert and Threlfall have the above construction in section 60 of their book, but only for 3-dimensional manifolds. The same works in all dimensions, just things become messier.
answered Dec 28 '18 at 1:53
Moishe CohenMoishe Cohen
46.1k342104
$endgroup$
This construction I am sure is written somewhere, most likely, in the book H.Seifert, W.Threlfall, "A Textbook of Topology" (I just do not have it with me now):
Let $M$ be a compact connected triangulated $n$-dimensional manifold (for simplicity, without boundary). Let $G$ be the graph dual to the triangulation $Delta$: The vertices of $G$ are the barycenters of the facets (n-dimensional simplices), two vertices are connected by an edge $e$ iff the corresponding facets have a common codimension 1 face denoted $e^*$ (a "panel"). Let $Tsubset G$ be a maximal subtree. Next, "cut open" $M$ along the panels $e^*$ for the edges $e$ of $G$ which are not in $T$. The result is a finite simplicial complex $Delta_T$. One then verifies that $T$ is isomorphic to a triangulated $n$-dimensional ball. With more work one proves that some subdivision of $Delta_T$ is isomorphic to a convex simplicial (i.e. its faces define a triangulated sphere) polyhedron in $R^n$. My suggestion is to work this out in the case of surfaces, say, of a triangulated 2-dimensional sphere.
Now, one can reverse this process and glue appropriate faces of $Delta_T$ (the ones which have the same image in $M$). The result is your triangulated manifold $M$.
Edit. I finally checked: Seifert and Threlfall have the above construction in section 60 of their book, but only for 3-dimensional manifolds. The same works in all dimensions, just things become messier.
answered Dec 28 '18 at 1:53
Moishe CohenMoishe Cohen
46.1k342104
edited Jan 7 at 1:26
answered Dec 28 '18 at 1:53
Moishe CohenMoishe Cohen
46.1k342104
answered Dec 28 '18 at 1:53
Moishe CohenMoishe Cohen
46.1k342104
answered Dec 28 '18 at 1:53
Moishe CohenMoishe Cohen
46.1k342104
46.1k342104
add a comment |
add a comment |
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1
$begingroup$
Surfaces are to a ridiculous but fantastic degree much simpler than higher dimensional manifolds. There is no hope to say anything reasonable about higher dimensional manifolds by the same techniques.
$endgroup$
– Mike Miller
Dec 20 '18 at 2:15
1
$begingroup$
There should be a class of complexes that we can build using this approach (?) For example, it seems to me that N-dimensional Torus and Sphere can be built this way.
$endgroup$
– self-educator
Dec 20 '18 at 3:21
1
$begingroup$
Your question as asked in un-answerable since you did not specify what cut and glue operations your are allowing. I suggest you first work out what your question is for discrete topological spaces (0-dimensional simplicial complexes), then for simplicial graphs. Also, keep in mind that for each connected n-dimensional triangulated manifold $M$, you can obtain $M$ by gluing some boundary faces of a triangulated $n$-dimensional ball. Maybe this is what you wanted to know.
$endgroup$
– Moishe Cohen
Dec 25 '18 at 22:51
$begingroup$
I'm interested in building n-dimensional simplicial complexes using n-dimensional polytopes + cut and glue (analogous to the role of cut and glue in building surfaces). In particular, I'm wondering what type of n-dimensional complexes we can/can't build, starting from an arbitrary n-dimensional polytope and gluing faces together.
$endgroup$
– self-educator
Dec 26 '18 at 5:36
1
$begingroup$
Still unclear. But it seems that you are allowing just one polytope for each space you are building. Then, as I said, you can get any compact connected triangulated manifold (possibly after subdividing the triangulation since you are insisting on polytopes). You can get many other complexes (already in dimension 2). There is no good description besides tautological one.
$endgroup$
– Moishe Cohen
Dec 27 '18 at 3:19