Throughout this talk, we will compare the notions of topological isotopy, smooth isotopy, and smooth equivalence (via an ambient diffeomorphism preserving homology) between homotopic $2$-spheres smoothly embedded in a $4$-manifold. In particular, we will construct pairs of spheres that are smoothly equivalent but not even topologically isotopic. Indeed, our examples show that Gabai's recent "4D Lightbulb Theorem" does not hold without the 2-torsion hypothesis.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/9096