Universal Realization of Avalanche Complexes on Directed Trees
Overview
In parallel chip-firing, sets of vertices firing together generate an avalanche complex. This paper realizes every finite simplicial complex with a vertex inside such a complex on a tree directed towards one absorbing sink, using only zero or one initial grain at each vertex. Explicit simplicial collapses remove the auxiliary vertices and preserve the target's homotopy type. The construction answers Riihimäki and Smith's Section 5, Question 2 for finite wedges of spheres and their torsion and non-wedge questions. Compatible retractions also realize nonempty finite filtrations after time refinement and sampling. The separate fixed-path Conjecture 3.11 is not addressed. The paper contains complete proofs, not yet formalized.
Original public questions
Riihimäki–Smith Question 2: finite realization of avalanche complexes
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.
Original abstract (English)
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.
Mathematical review
The manuscript has completed internal review. Complete formalization is not yet established for this public version.
Review standard