Point-particles moving in a background space are mathematically modelled by configurations spaces. Data associated to the particles such as spin are incorporated by giving the configurations labels in a suitable state space. These spaces have seen much attention in topology starting with work of McDuff and Segal in the 1970s. In classical field theory, however, point-particles interact with fields. Mathematically fields give rise to functions on the complement of a configuration, and thus to the configuration mapping spaces of the title.

Their systematic study has been initiated by Ellenberg, Venkatesh and Westerland motivated by their work on the Cohen-Lenstra conjecture for function spaces. Hurwitz spaces are homotopy equivalent to configuration mapping spaces on the 2-disc and their homology stability (as the number of points is increased) is a key step in their work.

In joint work with Martin Palmer we build on their work and prove analogous results for higher dimensional manifolds.