Martin Palmer-Anghel (University of Leeds) – Homological stability for asymptotic monopole moduli spaces

Magnetic monopoles were introduced by Dirac in 1931 to explain the quantisation of electric charges. In his model, they are singular solutions to an extension of Maxwell's equations allowing non-zero magnetic charges. An alternative model, developed by 't Hooft and Polyakov in the 1970s, is given, after a certain simplification, by smooth solutions to a different set of equations, the Bogomolny equations, whose moduli space of solutions has connected components M_k indexed by positive integers k (the "total magnetic charge"). These moduli spaces, which are non-compact manifolds, have an interpretation in terms of rational self-maps of CP¹ due to Donaldson and their stable homotopy types may be described in terms of braid groups by a result of F. Cohen, R. Cohen, Mann and Milgram. A partial compactification of M_k has recently been constructed by Kottke and Singer, whose boundary strata may be called "ideal" or "asymptotic" monopole moduli spaces. I will describe joint work with Ulrike Tillmann in which we prove the existence of stability patterns in the homology of these spaces.