Betti numbers of a persistence fiber with no finite bars
Let $K$ be a finite connected simplicial complex, take homology with coefficients in $\mathbb{F}_2$, and write $\beta_q(K) = \dim_{\mathbb{F}_2} H_q(K;\mathbb{F}_2)$. Let $D_K$ be the graded barcode with one infinite bar born at $1$ in degree zero, $\beta_q(K)$ infinite bars born at $2$ in each degree $q \ge 1$, and no finite bars. Let $m$ be the number of distinct finite endpoints of $D_K$ (so $m = 1$ if every positive-degree Betti number vanishes and $m = 2$ otherwise) and $I = [1, m+1]$. A filter is a map $f : K \to I$ on the nonempty simplices with $f(\tau) \le f(\sigma)$ whenever $\tau$ is a face of $\sigma$; its sublevel complexes are $K_t(f) = \{\sigma \in K : f(\sigma) \le t\}$. Let $\operatorname{PH}(f)$ be the graded barcode of the ordinary, unreduced persistent homology of these sublevel complexes, and give $$F_K = \{f : \operatorname{PH}(f) = D_K\}$$ the subspace topology of $I^{|K|}$, where $|K|$ is the number of nonempty simplices. Leygonie and Tillmann proved that for a simplicial complex the fiber of the persistence map over any barcode is a polyhedral complex, and Leygonie and Henselman-Petrusek computed such fibers with an algorithm; their computations over $\mathbb{F}_2$ led to this question.
Problem. Is $$\dim_{\mathbb{F}_2} H_q(F_K;\mathbb{F}_2) = \beta_q(K)$$ for every such $K$ and every $q \ge 0$? The filters range over all assignments to the simplices; restricting to filters determined by vertex values changes the question.
Source: Leygonie and Henselman-Petrusek (2024), Conjecture 1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AT — Algebraic Topology
- Secondary
- CO
- Keywords
- AIM Open Applied Problems, fibers of the persistence map, persistent homology
References
@article{leygonie2024,
author = {Leygonie, Jacob and Henselman-Petrusek, Gregory},
title = {Algorithmic reconstruction of the fiber of persistent homology on cell complexes},
journal = {J. Appl. Comput. Topol.},
volume = {8},
number = {7},
year = {2024},
pages = {2015--2049},
doi = {10.1007/s41468-024-00165-w},
eprint = {2110.14676},
archivePrefix = {arXiv},
}
@article{leygonie2022fiber,
author = {Leygonie, Jacob and Tillmann, Ulrike},
title = {The fiber of persistent homology for simplicial complexes},
journal = {J. Pure Appl. Algebra},
volume = {226},
number = {12},
year = {2022},
pages = {107099},
doi = {10.1016/j.jpaa.2022.107099},
eprint = {2104.01372},
archivePrefix = {arXiv},
}
@misc{aim-persistence-fiber-betti-numbers,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Betti numbers of a persistence fiber with no finite bars}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/537-persistence-fiber-betti-numbers.md},
note = {Entry 537 at commit 24edaad, 2026-10-15},
}
Resolution claims
For a finite connected simplicial complex, consider all simplex-wise filters whose ordinary persistent homology has one infinite degree-zero bar born at time 1 and all positive-degree bars born at time 2, with no finite bars. We prove that this fiber has the same homology over $\F$ as the original complex. The fiber has a deformation retract homeomorphic to the order …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/persistence_fiber.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00817, "Betti numbers of a persistence fiber with no finite bars", clopen.solutions.