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 $R^{\prime }$ 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 $R^{\prime }$ 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 $4$-dimensional light bulb theorem. We also give similar statements when $R$ and $R^{\prime }$ are orientable positive-genus surfaces.

20 pages, 10 figures

References in corpus (1)

• Concordances from the standard surface in $S^2\times S^2$