Differentiable classification of 4-manifolds with singular Riemannian foliations

Abstract

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: $$\mathbb {S}^4$$ , $$\mathbb {CP}^2$$ , $$\mathbb {S}^2\times \mathbb {S}^2$$ , or $$\mathbb {CP}^2$$ # $$\pm \mathbb {CP}^2$$ . 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 $$\mathbb {S}^4$$ , $$\pm \mathbb {CP}^2$$ 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.