Differentiable classification of 4-manifolds with singular Riemannian foliations

Abstract

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: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 S 4 , ±CP 2 and S 2 × 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.

Singular Riemannian foliations and applications to positive and non-negative curvature: Figure 1.

“…If M * has four vertices, then M * is isometric to a flat rectangle and M also admits a double disk bundle decomposition. In this case, it follows from Theorem 1.1 in [17] that M is diffeomorphic to one of…”

Positive curvature and symmetry in small dimensions

“…Furthermore, in dimension 4, the best result is due to Hsiang-Kleiner and Grove-Wilking (see [HK89,GW14], cf. [GGR15, GR15,PP03]), while in dimension 5, the best result is due to Rong and Galaz-Garcia-Searle (see [Ron02,GGS14], cf. [Goz15,Sim]).…”

Rational spheres and double disk bundles

“…First, if ℓ 2 = 2, then 1 + 2 = n − 1, so n = 4. As mentioned previously, it follows from [9] that M must be diffeomorphic to S 4 in this case. Hence, we assume ℓ 2 4.…”

“…The case n = 4 was previously shown by Ge and Radeschi [9] as a consequence of their classification of singular Riemannian foliations in dimension 4. The case n = 6 was already shown by the author, Galaz-García, and Kerin [5].…”

“…Hence, 2n − 1 = 7, so n = 4. As mentioned in the introduction, diffeomorphism rigidity follows from Ge and Radeschi's classification of four-dimensional singular Riemannian foliations [9].…”