What happens to topological invariants (and black holes) in singularity-free theories?

Potentials arising in ultraviolet-completed field theories can be devoid of singularities, and hence render spacetimes simply connected. This challenges the notion of topological invariants considered in such scenarios. We explore the classical implications for (i) electrodynamics in flat spacetime, (ii) ultrarelativistic gyratonic solutions of weak-field gravity, and (iii)the Reissner--Nordström black hole in general relativity. In linear theories, regularity spoils the character of topological invariants and leads to radius-dependent Aharonov--Bohm phases, which are potentially observable for large winding numbers. In general relativity, the physics is richer: The electromagnetic field can be regular and maintain its usual topological invariants, and the resulting geometry can be interpreted as a Reissner--Nordström black hole with a spacetime region of coordinate radius $\sim q^2/(GM)$ cut out. This guarantees the regularity of linear and quadratic curvature invariants ($\mathcal{R}$ and $\mathcal{R}^2$), but does not resolve singularities in invariants such as $\mathcal{R}^p\Box^n \mathcal{R}^q$, reflected by conical or solid angle defects. This motivates that gravitational models beyond general relativity need to be considered. These connections between regularity (= UV properties of field theories) and topological invariants (= IR observables) may hence present an intriguing avenue to search for traces of new physics and identify promising modified gravity theories.