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Integrable embeddings and foliations
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 134, Number 3, June 2012
- pp. 773-825
- 10.1353/ajm.2012.0018
- Article
- Additional Information
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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.