Timeline for Subdivision of simplicial sets, but not the barycentric one
Current License: CC BY-SA 4.0
6 events
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| Aug 9, 2021 at 19:12 | comment | added | Russ Woodroofe | I'd generally interpret being a subdivision as meaning "arising by repeated stellar subdivision." For example, the barycentric subdivision arises by iteratively subdividing all faces in a linear extension of reverse inclusion. OTOH, Stanley in Combinatorics and commutative algebra has an alternative that you might like more: he defines a subdivision of simplicial complex \Delta to be a complex \Delta' s.t. each face of \Delta' is contained in a face of \Delta, where his inclusion comes from a geometric realization. Should be easy to abstract out usefully, I think. | |
| Aug 7, 2021 at 5:20 | comment | added | Tim Porter | My original question was simply along the lines of 'What should be the general definition / meaning of `$K$ is a subdivision of $L$' and from there to get to a notion of 'finer' subdivision, and (not in the question) to be able to work with the system of all finer 'triangulations' of a given simplicial set. (I needed this, at the time, for working with triangulated cobordisms in Topological Quantum Field Theory, but the question was independent of the constraints of that context.) | |
| Aug 6, 2021 at 19:04 | comment | added | Russ Woodroofe | Are you looking to detect whether K is obtained from L by repeated subdivisions computationally? Or in what sense do you want to detect subdivisions? Another term that might be along the lines of what you are looking for is "bistellar flip", although this is not in general straightforwardly a subdivision. But I'll mention that Frank Lutz and others have done computational work on bistellar flips for manifolds. | |
| Aug 6, 2021 at 6:02 | comment | added | Tim Porter | Again your construction may help (I did know of that construction as it is clssical for simplicial complexes.) even for general simplicial sets, but would need to be restricted to the non-degenrate simplices, Thanks as it reminded me of things that needed to be recalled (by me)! | |
| Aug 6, 2021 at 6:01 | comment | added | Tim Porter | Although I agree that that is a good subdivision it does not quite answer my question. The point is rather to have two simplicial sets, $K$ and $L$ and we want to say that $K$ is a subdivision of $L$. Perhaps one might say that we need extra data, e.g. a monomorphism from $L_0$ to $K_0$ satisfying some conditions. You might be able to dream up some such conditions by iterating your construction, and I agree that the join operation should be in their somewhere but exactly how is not 100% clear. | |
| Aug 5, 2021 at 19:17 | history | answered | Russ Woodroofe | CC BY-SA 4.0 |