Timeline for Persistent homotopy groups
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| May 24, 2021 at 10:35 | comment | added | Greg Friedman | Not really an answer about barcodes, but see also: arxiv.org/abs/2002.10013 | |
| May 21, 2021 at 22:08 | comment | added | Andrea Marino | Ok, it seems like one of the main points is the difficulty in computations. Probably van kampen for fundamental group makes it computable. More formally, I am asking if there is a finite resolution in persistent groups of the persistent homotopy groups of a filtered space, where each factor of the resolution is a "free" barcode. With this I mean that the group is free at each $t$. This is a possible translation of the intuitive fact that the honotopy group can be presented parametrically in t. | |
| May 21, 2021 at 13:59 | comment | added | Noah Snyder | Right, but the Kenzo calculations are very slow even for very simple spaces. | |
| May 21, 2021 at 9:13 | comment | added | Dima Pasechnik | so, a natural question is - do homotopy groups of Rips complex carry interesting information (more than the homology groups do)? | |
| May 21, 2021 at 9:11 | comment | added | Dima Pasechnik | people do compute homotopy groups on computer, cf e.g. www-fourier.ujf-grenoble.fr/~sergerar/Kenzo | |
| May 21, 2021 at 8:15 | comment | added | Achim Krause | I think this "barcode" description relies on field coefficients. A sequence of abelian groups like $0\to \mathbb{Z} \to \mathbb{Z}/2\to \ldots$ certainly doesn't decompose into "bars" that just look like a single generator appearing and disappearing again. You can of course still do homological algebra of $[0,1]$-indexed sequences of abelian groups with finitely many critical values, and homotopy groups of a $[0,1]$-filtered space give an example, but it's unclear what exactly you're asking. | |
| May 21, 2021 at 7:51 | comment | added | Eric Peterson | I haven’t read this paper for myself, but you may be interested in ongoing work of Zhou: arxiv.org/abs/1912.12399 . | |
| May 21, 2021 at 6:59 | history | edited | YCor |
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| May 20, 2021 at 23:30 | comment | added | Ryan Budney | Of course there is a theory... but do you want it to have any useful properties? One can talk about the homotopy groups of a filtered complex. This was done before Persistent Homology was an object of study, with success. Your question seems rather vague. | |
| May 20, 2021 at 23:06 | comment | added | user164898 | Yes, homotopy groups are harder to compute. Homology is easy: given a finite point cloud, you can form its Vietoris-Rips complex, then take its simplicial chain complex. This is a finite chain complex of finite-dimensional vector spaces, very tractable for a computer, and you can use standard lin. alg. packages. But to algorithmically calculate the homotopy groups of the VR complex, you have to replace the simplicial set of the complex with a Kan-fibrant model, e.g. the Dwyer-Kan loop group or the Ex^infty construction. Both infinite, and unclear how to encode in a data type, for a computer. | |
| May 20, 2021 at 22:26 | history | edited | Andrea Marino | CC BY-SA 4.0 |
added 145 characters in body
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| May 20, 2021 at 22:12 | history | asked | Andrea Marino | CC BY-SA 4.0 |