Embedded 2-spheres K and J in the 4-sphere are 0-cobordant if they are joined in S^4 x I by an orientable 3-manifold M, such that each regular level set of M is a collection of 2-spheres. Ogasa introduced a local 0-cobordism called a ribbon move, and found several obstructions to ribbon move equivalence. In this talk we go from local to global by proving that K and J are 0-cobordant if and only if they are ribbon move equivalent. This means that Ogasa’s invariants obstruct 0-cobordance as well: in particular, there are infinitely many 0-cobordism classes of 2-knots. We also apply this theorem to the isotopy of unit spheres in CP^2.

This is joint work with Hannah Schwartz.