Concordance of surfaces in 4‐manifolds and the Freedman–Quinn invariant

Abstract

We prove a concordance version of the 4-dimensional light bulb theorem for $\pi_1$-negligible compact orientable surfaces, where there is a framed but not necessarily embedded dual sphere. That is, we show that if $F_0$ and $F_1$ are such surfaces in a 4-manifold $X$ that are homotopic and there exists an immersed framed 2-sphere $G$ in $X$ intersecting $F_0$ geometrically once, then $F_0$ and $F_1$ are concordant if and only if their Freedman-Quinn invariant $\mathop{fq}$ vanishes. The proof of the main result involves computing $\mathop{fq}$ in terms of intersections in the universal covering space and then applying work of Sunukjian in the simply-connected case.