Differentiable classification of 4-manifolds with singular Riemannian foliations

In this paper, we first prove that any closed simply connected 4-manifold that admits a decomposition into two disk bundles of rank greater than 1 is diffeomorphic to one of the standard elliptic 4-manifolds: ${\mathbb{S}}^4$, ${\mathbb{C}\mathbb{P}}^2$, ${\mathbb{S}}^2\times {\mathbb{S}}^2$, or $$\mathbb{C}\mathbb{P}$^2#\pm $\mathbb{C}\mathbb{P}$^2$. As an application we prove that any closed simply connected 4-manifold admitting a nontrivial singular Riemannian foliation is diffeomorphic to a connected sum of copies of standard ${\mathbb{S}}^4$, $\pm {\mathbb{C}\mathbb{P}}^2$ and ${\mathbb{S}}^2\times {\mathbb{S}}^2$. A classification of singular Riemannian foliations of codimension 1 on all closed simply connected 4-manifolds is obtained as a byproduct. In particular, there are exactly 3 non-homogeneous singular Riemannian foliations of codimension 1, complementing the list of cohomogeneity one 4-manifolds.