Tomasz Maciazek: Homology groups of graph configuration spaces
Time: Thu 2020-03-12 10.15 - 11.15
Participating: Tomasz Maciazek, Bristol
The n-particle configuration space of a topological space, X, is defined as the space of all configurations of n point-like particles in X with collision points excluded, modulo permutations of particles. Configuration spaces appear in physics in the contest of so-called quantum statistics. For instance, by considering configuration space of n particles in three dimensional real space, one can prove that spineless particles can have only two kinds of quantum statistics - bosonic and fermionic. More generally, physicists are often interested in determining isomorphism classes of flat hermitian vector bundles over certain configuration spaces. If particles are constrained to move on a graph (understood as a one-dimensional CW-complex), lots of information about isomorphism classes of such bundles can be extracted from homology groups via K-theory.
In the main part of my talk, I will review methods of computing homology groups of graph configuration spaces. In particular, I will describe discrete models for graph configuration spaces that were introduced by Abrams (2000) and Świątkowski (2001). As their applications, I will show how to compute homology groups of configuration spaces of trees (Abrams model) as well as wheel graphs and theta-graphs (Świątkowski model). In other words, I will present solutions to the universal presentation problem for homology groups of configuration spaces of the above families of graphs. This is based on my PhD work (2018) under the supervision of Adam Sawicki.