Department of Mathematics In 2017, Gabai proved the light bulb theorem, showing that if $R$ and $R'$ are 2-spheres homotopically embedded in a 4-manifold with a common dual, then with some condition on 2-torsion in $\pi_1(X)$ one can conclude that $R$ and $R'$ are smoothly isotopic. Schwartz later showed that this 2-torsion condition is necessary, and Schneiderman and Teichner then obstructed the isotopy whenever this condition fails. I showed that when $R'$ does not have a dual, we may still conclude the spheres are smoothly concordant.
I will talk about these various definitions and theorems as well as new joint work with Michael Klug generalizing the result on concordance to the situation where $R$ has an immersed dual (and $R'$ may have none), which is a common condition in 4-dimensional topology.