Smooth classification of four-dimensional rho-spheres

Establish whether every rho-sphere S^4_{\rho,\varepsilon} arising as a regular level set of an analytic control function defining the origin is smoothly diffeomorphic to the standard four-sphere, thereby resolving the remaining smooth four-dimensional case in the topology theorem for rho-spheres.

Background

The paper proves that all sufficiently small $\rho$-spheres are smoothly diffeomorphic to one another and that each is homeomorphic to the standard sphere. Using homological and h-cobordism arguments, it further obtains a smooth diffeomorphism with the standard sphere in all dimensions except when $m=5$.

When $m=5$, the $\rho$-sphere has dimension four, and the argument reduces to the unresolved smooth four-dimensional case. Resolving whether these four-dimensional homotopy spheres are smoothly standard would complete the smooth classification asserted by the theorem.