A concordance analogue of the $4$-dimensional light bulb theorem

Abstract

We prove a concordance analogue of Gabai's $4$-dimensional light bulb theorem. That is, we show that when $R$ and $R'$ are homotopically (smoothly) embedded $2$-spheres in a $4$-manifold $X^4$ where $\pi_1(X^4)$ has no $2$-torsion and one of $R$ or $R'$ has a transverse sphere, then $R$ and $R'$ are concordant. When $\pi_1(X^4)$ has $2$-torsion, we give a similar statement with extra hypotheses as in the $4$-dimensional light bulb theorem. We also give similar statements when $R$ and $R'$ are orientable positive-genus surfaces.