Timeline for Is the configuration space of ordered triples of distinct points in the four-edge banana graph homotopy equivalent to a surface of genus 13?
Current License: CC BY-SA 3.0
17 events
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| Sep 15, 2017 at 16:24 | answer | added | John Wiltshire-Gordon | timeline score: 4 | |
| Sep 15, 2017 at 14:59 | comment | added | j.c. | just adding a link to John Wiltshire-Gordon's preprint discussing the explicit simplicial model mentioned in the question arxiv.org/abs/1706.06626 | |
| May 8, 2017 at 20:58 | answer | added | Gabriel C. Drummond-Cole | timeline score: 10 | |
| May 8, 2017 at 18:56 | history | edited | John Wiltshire-Gordon | CC BY-SA 3.0 |
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| May 8, 2017 at 18:46 | comment | added | John Wiltshire-Gordon | @j.c. Ah, I misunderstood this distinction. (I am only interested in the "topological" configuration spaces) . I will edit accordingly. | |
| May 8, 2017 at 17:35 | comment | added | j.c. | @RyanBudney I guess you mean that the fundamental group has to be that of a 2-manifold? Then (to John Wiltshire-Gordon) is the 120 vertex complex indeed a triangulation of $\Sigma_{13}$, or am I missing something? | |
| May 8, 2017 at 17:21 | comment | added | j.c. | I'm having trouble extracting the results you cite for $C_3(K_5)$ and $C_4(K_{3,3})$ from Abrams's thesis. He distinguishes the standard "topological" $n$-point configuration space $C^{top}_n(G)$ from the "combinatorial configuration space", a subcomplex of $G^n$ that he calls $C_n(G)$ and proves in Theorem 2.4 that $C^{top}_2(G)$ deformation retracts to $C_2(G)$ for $G$ a simple graph (Theorem 2.1 gives some other conditions). Chapter 5 discusses the combinatorial configuration spaces of the graphs you cite, but I do not see a homotopy equivalence to $C^{top}_n$ for the last two cases. | |
| May 6, 2017 at 2:41 | comment | added | W. Cadegan-Schlieper | If you could show the higher homotopy groups to be zero, that would be sufficient. | |
| May 5, 2017 at 21:16 | comment | added | Ryan Budney | It sounds like it. Although I have not read Benedetti and Lutz's paper I imagine their moves maintain the simple homotopy type of an object, so it maintains homotopy type. A 2-dimensional pseudomanifold whose fundamental group is that of a manifold has to be a manifold (well, provided it does not have boundary). . . So that should do it. | |
| May 5, 2017 at 20:57 | comment | added | John Wiltshire-Gordon | @RyanBudney Using the collapsed complex, I have computed the fundamental group, which matches the fundamental group of $\Sigma_{13}$. Is this enough to conclude a homotopy equivalence? | |
| May 5, 2017 at 20:54 | history | edited | John Wiltshire-Gordon | CC BY-SA 3.0 |
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| May 4, 2017 at 21:48 | comment | added | Ryan Budney | If there's no 3-simplices, perhaps the least-energy thing you could do is collapse a maximal forest and feed the group presentation to some code that does small cancellation theory. The software Heegaard is very powerful in this regard. The group presentation code in Regina (mostly written by me) is quite capable but often Heegaard is more powerful. If you can find it, the software Magnus (available for some linux systems) is very versatile and powerful. There is probably other sortware out there, but that's where I would start. | |
| May 4, 2017 at 20:30 | comment | added | John Wiltshire-Gordon | Looks like there are no 3-simplices or higher. So I have a finitely presented groupoid with 336 objects, 1200 generating arrows, and 840 relations of the form x*y = z. | |
| May 4, 2017 at 20:20 | comment | added | Ryan Budney | Off the top of my head I'm not certain. I imagine you'd want to have a package to manipulate arbitrary (mostly) 2-dimensional CW-complexes, which is pretty much equivalent to groupoid presentations. Your cell complexes, for these configuration spaces, do they have any 3-cells or have they had those simplified-away? | |
| May 4, 2017 at 19:56 | comment | added | John Wiltshire-Gordon | @RyanBudney Good idea! It would be helpful for other things as well to has a smaller model. Any suggestions about what package would be easy to use? | |
| May 4, 2017 at 19:54 | comment | added | Ryan Budney | Does your simplicial complex have any free faces? If so, put it into a software package that does basic simple-homotopy equivalences / Whitehead moves and try some greedy simplifications. | |
| May 4, 2017 at 19:44 | history | asked | John Wiltshire-Gordon | CC BY-SA 3.0 |