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- W2058237977 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 $Phi:Mrightarrow {Bbb{E}}^{n-k}$ such that $Lsubset Phi^{-1}(0)$ [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." @default.
- W2058237977 created "2016-06-24" @default.
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- W2058237977 date "2012-01-01" @default.
- W2058237977 modified "2023-10-17" @default.
- W2058237977 title "Integrable embeddings and foliations" @default.
- W2058237977 cites W1535022931 @default.
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- W2058237977 cites W1943406019 @default.
- W2058237977 cites W1965989024 @default.
- W2058237977 cites W1970702640 @default.
- W2058237977 cites W1971927646 @default.
- W2058237977 cites W1973768104 @default.
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- W2058237977 cites W2005004957 @default.
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- W2058237977 cites W2024174238 @default.
- W2058237977 cites W2028633121 @default.
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- W2058237977 cites W2036122864 @default.
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- W2058237977 cites W2044034372 @default.
- W2058237977 cites W2045016477 @default.
- W2058237977 cites W2045055308 @default.
- W2058237977 cites W2045632804 @default.
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- W2058237977 cites W2075831638 @default.
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- W2058237977 doi "https://doi.org/10.1353/ajm.2012.0018" @default.
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