Concordance of spheres in 4-manifolds with an immersed dual sphere

Let $S_0$ and $S_1$ be two homotopic, oriented 2-spheres embedded in an orientable 4-manifold $X$. After discussing several operations for modifying an immersion of a 3-manifold into a 5-manifold, we discuss the Freedman–Quinn (fq) and Stong (stong) concordance obstructions. When these are defined for the pair $S_0,S_1$, they are defined in terms of the self-intersection set of a regular homotopy from $S_0$ to $S_1$. When $S_0$ has an immersed dual sphere, we see that under some mild topological conditions on $X$, the invariants fq and stong are a complete set of concordance obstructions. This work is an adaption of the methods of Richard Stong to the context of concordances of 2-spheres.