Timeline for Is my combinatorial set a CW complex?
Current License: CC BY-SA 4.0
13 events
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| Jun 28, 2019 at 17:58 | comment | added | Dima Pasechnik | the standard way is called barycentric subdivision. e.g. add an extra vertex in the center of an n-gon and connect it to all the other vertices. topologically the latter is still a disk, one has not punched any holes in it (and similarly in general) | |
| Jun 28, 2019 at 13:36 | comment | added | Viviane | Dima, I don't quite understand how the triangulation can have the same topology. But anyway, I don't know how to triangulate my faces! | |
| Jun 28, 2019 at 13:33 | comment | added | Viviane | Thank you Hugh for your remark. Indeed, you are right! In my case, every face is actually a simple polytope which translates into some combinatorial property by looking at the ideals of my face-inclusion poset. If I add this property as a hypothesis, then it forbids the missing edge triangle. Anyway, as you say, I am looking for the combinatorial abstraction of a polytopal complex. I guess if does not exist, it is an interesting question to look at. | |
| Jun 28, 2019 at 12:19 | comment | added | Dima Pasechnik | IMHO one can always triangulate polytopes involved, so that you get.a simplicial complex with the same topology as the original polyhedral complex. | |
| Jun 28, 2019 at 11:25 | comment | added | Hugh Thomas | It seems like your definition would include a triangle with one edge removed (but with all three of its vertices). This seems kind of pathological (and in particular wouldn't be a CW complex). I am not sure how to rule out such behaviour by a purely combinatorial criterion, though. It seems like you are looking for a combinatorial abstraction of polyhedral complexes, and I am not sure there is any good one. | |
| Jun 28, 2019 at 7:14 | history | edited | Viviane | CC BY-SA 4.0 |
added 700 characters in body
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| Jun 27, 2019 at 23:54 | comment | added | Dima Pasechnik | topologically it is often OK to triangulate your faces, cells, etc, just as triangulating faces of a planar graph does not break planarity (the latter would allow one to pass from general polytopes in $\mathbb{R}^3$ to simplicial ones) | |
| Jun 27, 2019 at 21:45 | comment | added | Viviane | No it's not, this I am sure. It has the same combinatorial properties though expect it's not made of triangle... | |
| Jun 27, 2019 at 20:02 | comment | added | Dima Pasechnik | is it a simplicial complex, perhaps? (these are a particular type of CW-complexes, much easier to work with) | |
| Jun 27, 2019 at 15:34 | comment | added | Viviane | That's what I thought. I am having trouble understanding how this topological properties translates in terms of combinatorics. I will probably end up proving that this is a polytopal complex, which solves the problem. But I also wonder if the combinatorial object I describe has a name. I call it a "combinatorial complex" because any other way to refer to it... | |
| Jun 27, 2019 at 15:05 | comment | added | Lee Mosher | You definitely need to prove more. An $n$-dimensional face of a CW complex is not just an abstract set of $0$-cells, it is an actual $n$-dimensional cell, in particular its interior is homeomorphic to an actual open ball in $n$-dimensional Euclidean space. Also, the face relation between cells is not just an abstract inclusion of sets of $0$-cells, it is an actual topological relation between an $n$-dimensional cell and a lower dimensional cell. | |
| Jun 27, 2019 at 15:00 | review | First posts | |||
| Jun 27, 2019 at 15:12 | |||||
| Jun 27, 2019 at 14:58 | history | asked | Viviane | CC BY-SA 4.0 |