The fiber of persistent homology for simplicial complexes
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Authors
Leygonie, Jacob
Tillmann, Ulrike
Publication Date
2022-12Journal Title
Journal of Pure and Applied Algebra
ISSN
0022-4049
Publisher
Elsevier BV
Type
Article
This Version
AM
Metadata
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Leygonie, J., & Tillmann, U. (2022). The fiber of persistent homology for simplicial complexes. Journal of Pure and Applied Algebra https://doi.org/10.1016/j.jpaa.2022.107099
Abstract
We study the inverse problem for persistent homology: For a fixed simplicial complex K, we analyse the fiber of the continuous map PH on the space of filters that assigns to a filter f : K Ñ R the total barcode of its associated sublevel set filtration of K. We find that PH is best understood as a map of stratified spaces. Over each stratum of the barcode space the map PH restricts to a (trivial) fiber bundle with fiber a polyhedral complex. Amongst other we derive a bound for the dimension of the fiber depending on the number of distinct endpoints in the barcode. Furthermore, taking the inverse image PH ́1 can be extended to a monodromy functor on the (entrance path) category of barcodes. We demonstrate our theory on the example of the simplicial triangle giving a complete description of all fibers and monodromy maps. This example is rich enough to have a Möbius band as one of its fibers.
Embargo Lift Date
2023-04-21
Identifiers
External DOI: https://doi.org/10.1016/j.jpaa.2022.107099
This record's URL: https://www.repository.cam.ac.uk/handle/1810/336112
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International
Licence URL: https://creativecommons.org/licenses/by-nc-nd/4.0/
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