Homotopy versus isotopy: 2-spheres in 5-manifolds

(Joint with Danica Kosanović and Peter Teichner)

A curious consequence of the smooth 4-dimensional Light Bulb Theorem is that homotopic 2-spheres in a 4-manifold M with a common geometric dual are isotopic in M if and only if they are isotopic in the product of M with an interval! Geometric duals are useful in 4-dimensions for removing singularities in Whitney disks, and this talk will describe how the `general homotopy versus isotopy' problem for 2-spheres in 5-manifolds has a completely algebraic solution, thanks to the availability of level-preserving successful Whitney moves for converting a homotopy with vanishing intersection invariants to an isotopy. A different approach via homotopy groups of mapping spaces
using results of J-P Dax gives the same results, and both approaches work more generally for n-spheres in (2n+1)-manifolds.