有向树上雪崩复形的普适实现
Universal Realization of Avalanche Complexes on Directed Trees
研究概述
在并行筹码触发过程中，同时触发的顶点集合生成雪崩复形。本文从每个顶点初始仅有零或一粒筹码的构型出发，在朝向唯一吸收汇点的树上，将任意有顶点的有限单纯复形实现为雪崩复形的诱导子复形，并通过显式单纯塌缩去除辅助顶点、保留目标的同伦型。这解答了 Riihimäki–Smith 第 5 节问题 2 的有限球面楔和实现问题，以及随后提出的挠率与非楔和问题。相容回缩还可在细化时间并采样后实现非空有限滤过。另列的固定路径猜想 3.11 不在解答范围内。论文包含完整证明，尚未形式化。
原始公开问题
Riihimäki–Smith 问题 2：雪崩复形的有限实现
Riihimäki and Smith: Avalanche homology of digraphs via sandpile dynamics — Section 5, Question 2 and following torsion/non-wedge questions; finite digraph category. The separate fixed-path Conjecture 3.11 is not addressed.
原文摘要（英文）
The avalanche complex of a parallel chip-firing process is generated by its sets of simultaneously firing vertices. We prove that every finite simplicial complex with a vertex is a simplicial collapse of an avalanche complex on a finite tree directed towards a single absorbing sink, starting from a binary configuration. The construction prescribes a sequence of firing sets on distinguished vertices by sending grains along private delay paths. Each auxiliary vertex fires only once and can be removed without changing the prescribed complex. The resulting retractions commute with the time filtration, giving a realization theorem for finite filtrations after refinement of time. This answers the finite realization question of Riihimäki and Smith and their questions about torsion and non-wedge homotopy types. We give explicit size bounds and an example with first integral homology Z/2Z.
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