Concordance of Surfaces and the Freedman-Quinn Invariant

We prove a concordance version of the 4-dimensional light bulb theorem allowing for the dual sphere to be immersed with an arbitrary intersection number. That is, we show that if $S_0$ and $S_1$ are 2-spheres in a 4-manifold $X$ that are homotopic and there exists an immersed 2-sphere $G$ in $X$ intersecting $S_0$ geometrically once, then $S_0$ and $S_1$ are concordant if and only if their Freedman-Quinn invariant fq vanishes. This is an invariant of a pair of based-homotopic 2-spheres recently studied by Schneiderman and Teichner; we make sense of the invariant for freely homotopic spheres in this setting. The proof involves redefining fq in terms of covering spaces and then applying work of Sunukjian in the setting of simply-connected manifolds. We give a similar statement for higher genus surfaces whose fundamental groups map trivially into that of $X$.