Timeline for If $X = X_1 \cup \cdots \cup X_n$ is shellable, then is $(X_1 \cup \cdots \cup X_k)\cap X_{k+1}$ shellable?
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| Nov 1, 2022 at 14:29 | comment | added | Richard Stanley | For simplicial complexes the question is uninteresting, but for polyhedral complexes it makes more sense. The answer boils down to how one defines shellability. Usually the condition in Question1 is part of the definition. E.g., see Definition 2.3 at mi.fu-berlin.de/math/groups/discgeom/ziegler/Preprintfiles/…. | |
| Oct 29, 2022 at 18:31 | answer | added | Tim Campion | timeline score: 1 | |
| Oct 29, 2022 at 17:09 | comment | added | Wlod AA | Tim, thank you. Sometimes one gets some kind of a block, looks at something and doesn't see it (e.g. that a simplex is a simplex), it happens (to me too, of course). I feel that I shouldn't call my comment an Answer. I hope that now you can go forward in your research smoothly for a while. Best regards. | |
| Oct 29, 2022 at 14:58 | comment | added | Tim Campion | @WlodAA Ah, thank you! I see now that the question is much simpler than I had appreciated. By definition of shellability, $(X_1 \cup \cdots \cup X_k) \cap X_{k+1} \subseteq X_{k+1} = \Delta^d$ is the union of some nonempty subset of the codimension-1 faces $\Delta^{d-1}$. To see that this is shellable (in any ordering), it suffices to observe that if $Y,Z \subseteq \Delta^d$ are any two codimension-1 faces, then $Y \cap Z$ is a simplex of dimension $d-2$. If you were to write this as answer, I would gladly accept it! | |
| Oct 29, 2022 at 8:26 | comment | added | Wlod AA | The given intersection is a pure complex of codimension 1 of simplex $X_{k+1}$ hence it must be shellable. | |
| Oct 28, 2022 at 22:44 | history | asked | Tim Campion | CC BY-SA 4.0 |