Concordance of Surfaces 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 $\mathrm{fq}$ vanishes. The proof of the main result involves computing $\mathrm{fq}$ in terms of intersections in the universal covering space and then applying work of Sunukjian in the simply-connected case.

33 pages, 18 figures. We added a hypothesis to the main theorem that the dual sphere is framed; Example 7.2 illustrates the necessity of this hypothesis by constructing a smooth homotopy of spheres with nontrivial Kervaire-Milnor (Stong) invariant. We greatly improved the discussion of framings (Section 5)