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Abstract

A $k$-submanifold $L$ of an open $n$-manifold $M$ is called {\it weakly integrable (WI)} [resp. {\it strongly integrable (SI)}] if there exists a submersion $\Phi:M\rightarrow {\Bbb{E}}^{n-k}$ 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. ({\it 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 ({\it Indagationes Math.} 1983) asserting that any open orientable surface is SI in ${\Bbb{E}}^3$, and Watanabe's and Miyoshi's theorems ({\it Topology} 1993 and 1995) claiming that any link is WI in an open $3$-manifold. In the case $M={\Bbb{E}}^n$ 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 ({\it Topology} 1995) which states that any link in ${\Bbb{E}}^3$ 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 ${\Bbb{E}}^3$ 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 ${\Bbb{E}}^n$ which can be realized as proper leaves of foliations; we prove that ${\Bbb{S}}^3$ can be realized as a leaf of a foliation of ${\Bbb{E}}^n$, $n \geq 7$, but not in ${\Bbb{E}}^5$ or ${\Bbb{E}}^6$, which partially answers a question by Vogt ({\it 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 ${\Bbb{E}}^4$.

The theory of WI and SI submanifolds is a framework where many classical tools of differential and algebraic topology play a prominent role: Phillips-Gromov h-principle, Hirsh-Smale theory, complete intersections, Seifert manifolds, the theory of immersions and embeddings, obstruction theory, etc.

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