Explicit cocycles for graph Hall algebras VERY GOOD
For the reduced simplicial set whose simplices are subgraphs of a finite labelled undirected multigraph equipped with ordered partitions of their vertices, we construct explicit mod-two cohomology classes indexed by subgraphs. Vertex restriction pulls a supported top-dimensional cochain into a global cocycle. Permutation cycles pair with these classes as a Kronecker basis, while a vertex filtration bounds the total cohomology dimension using augmented Boolean order complexes. Alexander-Whitney multiplication recovers the graph Hall product. This gives a candidate proof of the finite-graph Hall/cohomology identification stated as Conjecture 1.1 by Toury. The construction concerns labelled subgraphs, not isomorphism classes of rooted forests. Loops and labelled parallel edges are allowed. Exact finite checks accompany the proof; priority and external mathematical assessment remain pending.
Working paper. Citable, but not yet screened for the AIRR collection.