Integrable embeddings and foliations

TL;DR

This paper characterizes weakly and strongly integrable submanifolds in open manifolds, providing explicit conditions, counterexamples, and applications to foliation theory, thereby advancing understanding of submanifold integrability and foliation leaves.

Contribution

It offers explicit criteria for WI and SI submanifolds in open manifolds, including higher codimension cases, and applies these results to classify foliation leaves and construct novel examples.

Findings

Full characterization of WI and SI submanifolds in R^n.

Counterexamples showing not all links or knots are SI.

Existence of open 3-manifolds that are leaves in R^4 but not in R^5 or R^6.

Abstract

A k-submanifold L of an open n-manifold M is called weakly integrable (WI) [resp. strongly integrable (SI)] if there exists a submersion \Phi:M\to R^{n-k} such that L\subset \Phi^{-1}(0) [resp. L= \Phi^{-1}(0)]. In this work we study the following problem, first stated in a particular case by Costa et al. (Invent. Math. 1988): which submanifolds L of an open manifold M are WI or SI? For general M, we explicitly solve the case k=n-1 and provide necessary and sufficient conditions for submanifolds to be WI and SI in higher codimension. As particular cases we recover the theorem of Bouma and Hector (Indagationes Math. 1983) asserting that any open orientable surface is SI in R^3, and Watanabe's and Miyoshi's theorems (Topology 1993 and 1995) claiming that any link is WI in an open 3-manifold. In the case M=R^n we fully characterize WI and SI submanifolds, we provide examples of 3- and 7-manifolds which are not WI and we show that a theorem by Miyoshi (Topology 1995) which states that any link in R^3 is SI does not hold in general. The right analogue to Miyoshi's theorem is also proved, implying in particular the surprising result that no knot in R^3 is SI. Our results applied to the theory of foliations of Euclidean spaces give rise to some striking corollaries: using some topological invariants we classify all the submanifolds of R^n which can be realized as proper leaves of foliations; we prove that S^3 can be realized as a leaf of a foliation of R^n, n\geq 7, but not in R^5 or R^6, which partially answers a question by Vogt (Math. Ann. 1993); we construct open 3-manifolds which cannot be leaves of a foliation of any compact 4-manifold but are proper leaves in R^4. The theory of WI and SI submanifolds is a framework where many classical tools of differential and algebraic topology play a prominent role: h-principle, complete intersections and the theory of immersions and embeddings.