A Concordance Analogue of the 4D Light Bulb Theorem

Abstract

Abstract We prove a concordance analogue of Gabai’s $4$D light bulb theorem. That is, we show that when $R$ and $R^{\prime}$ are homotopic, embedded $2$-spheres in a $4$-manifold $X^4,$ where $\pi _1(X^4)$ has no $2$-torsion and one of $R$ or $R^{\prime}$ has a transverse sphere, then $R$ and $R^{\prime}$ are concordant. When $\pi _1(X^4)$ has $2$-torsion, we give a similar statement with extra hypotheses as in the $4$D light bulb theorem. We also give similar statements when $R$ and $R^{\prime}$ are orientable positive-genus surfaces.