Stable and unstable homology of graph braid groups

Abstract: The homology of the configuration spaces of a graph forms a finitely generated module over the polynomial ring generated by its edges; in particular, each Betti number is eventually equal to a polynomial in the number of particles, an analogue of classical homological stability. The degree of this polynomial is captured by a connectivity invariant of the graph, and its leading coefficient may be computed explicitly in terms of cut counts and vertex valences. This "stable" (asymptotic) homology is generated entirely by the fundamental classes of certain tori of geometric origin, but exotic non-toric classes abound unstably. These mysterious classes are intimately tied to questions about generation and torsion whose answers remain elusive except in a few special cases. This talk represents joint work with Byung Hee An and Gabriel Drummond-Cole.

Mathematics

Audience: researchers in the discipline

Caltech geometry/topology seminar