Zero in the upper-Laplacian spectrum of low-degree simplicial complexes
Let $d \ge 2$ and let $X$ be a nonempty pure $d$-dimensional simplicial complex, possibly infinite; no finiteness is assumed of the degrees of faces of dimension below $d - 1$. For a $(d-1)$-simplex $\sigma$ let $\deg(\sigma)$ be the number of $d$-simplices containing it, and put $$D = \sup_{\sigma \in X^{d-1}} \deg(\sigma).$$ Fix orientations, give $(d-1)$-cochains the norm $$\|f\|^2 = \sum_{\sigma \in X^{d-1}} \deg(\sigma) |f(\sigma)|^2$$ and $d$-cochains the counting norm. The usual simplicial coboundary $$(\delta f)([v_0, \ldots, v_d]) = \sum_{i=0}^{d} (-1)^i f([v_0, \ldots, \widehat{v_i}, \ldots, v_d])$$ is then a bounded operator, and $\Delta^+ = \delta^* \delta$ is the normalized upper Laplacian on $(d-1)$-cochains; $A_0 = I - \Delta^+$ is the operator that governs Rosenthal's simplicial branching random walks. For the $k$-regular arboreal $d$-complex, Rosenthal computed the spectral measure of $A_0$: its spectrum is the interval $$I_{d,k} = \Bigl[\tfrac{1 - d - 2\sqrt{(k-1)d}}{k},\ \tfrac{1 - d + 2\sqrt{(k-1)d}}{k}\Bigr]$$ for $k \ge d + 1$, whose right endpoint is $1$ when $k = d + 1$, and $I_{d,k} \cup \{1\}$, with $1$ an eigenvalue, for $k < d + 1$.
Problem. Prove or disprove the following two assertions, with the spectrum and kernel taken on the full Hilbert space of $(d-1)$-cochains: (1) if $D \le d + 1$, then $0 \in \operatorname{Spec}(\Delta^+)$; (2) if $D \le d$, then $\ker \Delta^+ \ne \{0\}$, that is, zero is an eigenvalue with a nonzero square-summable eigencochain. Equivalently, $1 \in \operatorname{Spec}(A_0)$ when $D \le d + 1$, and $1$ is an eigenvalue of $A_0$ when $D \le d$.
Source: Rosenthal (2024), Conjecture 4.2 as AIM cites it (Conjecture 4.3 in arXiv:1412.5406v1: one is an eigenvalue of A_0 when D ≤ d). Stated by AIM — Open Applied Problems as two assertions, the first for D ≤ d + 1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AT, CO
- Keywords
- AIM Open Applied Problems, branching random walks, simplicial complexes, upper Laplacian
References
@article{rosenthal2024,
author = {Rosenthal, Ron},
title = {Simplicial branching random walks},
journal = {J. Appl. Comput. Topol.},
volume = {8},
number = {6},
year = {2024},
pages = {1751--1791},
doi = {10.1007/s41468-023-00148-3},
eprint = {1412.5406},
archivePrefix = {arXiv},
}
@misc{aim-low-degree-simplicial-zero-spectrum,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Zero in the upper-Laplacian spectrum of low-degree simplicial complexes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/643-low-degree-simplicial-zero-spectrum.md},
note = {Entry 643 at commit 24edaad, 2026-10-15},
}
Resolution claims
Does a low codimension-one degree force zero into the spectrum of the normalized upper Laplacian of an infinite simplicial complex? We construct a countable pure three-dimensional complex with maximum triangle degree three whose upper Laplacian has spectrum bounded away from zero on the full degree-weighted triangle cochain space. This disproves both assertions in AIM Problem 643, including the stronger claim …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/degree_three_counterexample.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00807, "Zero in the upper-Laplacian spectrum of low-degree simplicial complexes", clopen.solutions.