Examples of odd-dimensional manifolds that do not admit contact structure
$begingroup$
I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure.
Can someone provide me with some examples?
dg.differential-geometry at.algebraic-topology differential-topology contact-geometry
edited 1 hour ago
Piotr Hajlasz
9,46843672
asked 3 hours ago
Warlock of Firetop MountainWarlock of Firetop Mountain
24717
$endgroup$
add a comment |
$begingroup$
I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure.
Can someone provide me with some examples?
dg.differential-geometry at.algebraic-topology differential-topology contact-geometry
edited 1 hour ago
Piotr Hajlasz
9,46843672
asked 3 hours ago
Warlock of Firetop MountainWarlock of Firetop Mountain
24717
$endgroup$
1
$begingroup$
Not exactly answering the question since I'm not providing examples, but for a manifold $M^{2n+1}$, it turns out that admitting a contact structure is equivalent to admitting a reduction of structure group to $U(n) times 1$ (such a reduction is called an almost contact structure), as proved by Borman, Eliashberg, and Murphy. The contact structures they produce for a given almost contact class are overtwisted, meaning they contain some model overtwisted chart. It is a more difficult question to ask when a manifold admits a tight (= non-overtwisted) contact structure.
$endgroup$
– KSackel
2 hours ago
add a comment |
$begingroup$
I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure.
Can someone provide me with some examples?
dg.differential-geometry at.algebraic-topology differential-topology contact-geometry
edited 1 hour ago
Piotr Hajlasz
9,46843672
asked 3 hours ago
Warlock of Firetop MountainWarlock of Firetop Mountain
24717
$endgroup$
I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure.
Can someone provide me with some examples?
dg.differential-geometry at.algebraic-topology differential-topology contact-geometry
dg.differential-geometry at.algebraic-topology differential-topology contact-geometry
edited 1 hour ago
Piotr Hajlasz
9,46843672
asked 3 hours ago
Warlock of Firetop MountainWarlock of Firetop Mountain
24717
edited 1 hour ago
Piotr Hajlasz
9,46843672
asked 3 hours ago
Warlock of Firetop MountainWarlock of Firetop Mountain
24717
edited 1 hour ago
Piotr Hajlasz
9,46843672
edited 1 hour ago
Piotr Hajlasz
9,46843672
edited 1 hour ago
Piotr Hajlasz
9,46843672
9,46843672
asked 3 hours ago
Warlock of Firetop MountainWarlock of Firetop Mountain
24717
asked 3 hours ago
Warlock of Firetop MountainWarlock of Firetop Mountain
24717
asked 3 hours ago
Warlock of Firetop MountainWarlock of Firetop Mountain
24717
24717
1
$begingroup$
Not exactly answering the question since I'm not providing examples, but for a manifold $M^{2n+1}$, it turns out that admitting a contact structure is equivalent to admitting a reduction of structure group to $U(n) times 1$ (such a reduction is called an almost contact structure), as proved by Borman, Eliashberg, and Murphy. The contact structures they produce for a given almost contact class are overtwisted, meaning they contain some model overtwisted chart. It is a more difficult question to ask when a manifold admits a tight (= non-overtwisted) contact structure.
$endgroup$
– KSackel
2 hours ago
add a comment |
1
$begingroup$
Not exactly answering the question since I'm not providing examples, but for a manifold $M^{2n+1}$, it turns out that admitting a contact structure is equivalent to admitting a reduction of structure group to $U(n) times 1$ (such a reduction is called an almost contact structure), as proved by Borman, Eliashberg, and Murphy. The contact structures they produce for a given almost contact class are overtwisted, meaning they contain some model overtwisted chart. It is a more difficult question to ask when a manifold admits a tight (= non-overtwisted) contact structure.
$endgroup$
– KSackel
2 hours ago
1
1
$begingroup$
Not exactly answering the question since I'm not providing examples, but for a manifold $M^{2n+1}$, it turns out that admitting a contact structure is equivalent to admitting a reduction of structure group to $U(n) times 1$ (such a reduction is called an almost contact structure), as proved by Borman, Eliashberg, and Murphy. The contact structures they produce for a given almost contact class are overtwisted, meaning they contain some model overtwisted chart. It is a more difficult question to ask when a manifold admits a tight (= non-overtwisted) contact structure.
$endgroup$
– KSackel
2 hours ago
$begingroup$
Not exactly answering the question since I'm not providing examples, but for a manifold $M^{2n+1}$, it turns out that admitting a contact structure is equivalent to admitting a reduction of structure group to $U(n) times 1$ (such a reduction is called an almost contact structure), as proved by Borman, Eliashberg, and Murphy. The contact structures they produce for a given almost contact class are overtwisted, meaning they contain some model overtwisted chart. It is a more difficult question to ask when a manifold admits a tight (= non-overtwisted) contact structure.
$endgroup$
– KSackel
2 hours ago
add a comment |
1 Answer
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$begingroup$
Every compact orientable $3$-dimensional manifold has a contact structure [1]. On the other hand we have
Theorem. For $ngeq 2$ there is a closed oriented connected manifold of dimension $2n+1$ without a contact structure.
For $n=2$, $SU(3)/SO(3)$ has no contact structure and for $n>2$,
$SU(3)/SO(3)timesmathbb{S}^{2n-4}$ has no contact structure, see Proposition 2.4 in [2].
[1] J. Martinet,
Formes de contact sur les variétés de dimension 3. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp. 142–163. Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971.
[2] R. E. Stong, Contact manifolds. J. Differential Geometry 9 (1974), 219–238.
answered 2 hours ago
Piotr HajlaszPiotr Hajlasz
9,46843672
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add a comment |
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1 Answer
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1 Answer
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oldest
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$begingroup$
Every compact orientable $3$-dimensional manifold has a contact structure [1]. On the other hand we have
Theorem. For $ngeq 2$ there is a closed oriented connected manifold of dimension $2n+1$ without a contact structure.
For $n=2$, $SU(3)/SO(3)$ has no contact structure and for $n>2$,
$SU(3)/SO(3)timesmathbb{S}^{2n-4}$ has no contact structure, see Proposition 2.4 in [2].
[1] J. Martinet,
Formes de contact sur les variétés de dimension 3. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp. 142–163. Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971.
[2] R. E. Stong, Contact manifolds. J. Differential Geometry 9 (1974), 219–238.
answered 2 hours ago
Piotr HajlaszPiotr Hajlasz
9,46843672
$endgroup$
add a comment |
$begingroup$
Every compact orientable $3$-dimensional manifold has a contact structure [1]. On the other hand we have
Theorem. For $ngeq 2$ there is a closed oriented connected manifold of dimension $2n+1$ without a contact structure.
For $n=2$, $SU(3)/SO(3)$ has no contact structure and for $n>2$,
$SU(3)/SO(3)timesmathbb{S}^{2n-4}$ has no contact structure, see Proposition 2.4 in [2].
[1] J. Martinet,
Formes de contact sur les variétés de dimension 3. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp. 142–163. Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971.
[2] R. E. Stong, Contact manifolds. J. Differential Geometry 9 (1974), 219–238.
answered 2 hours ago
Piotr HajlaszPiotr Hajlasz
9,46843672
$endgroup$
add a comment |
$begingroup$
Every compact orientable $3$-dimensional manifold has a contact structure [1]. On the other hand we have
Theorem. For $ngeq 2$ there is a closed oriented connected manifold of dimension $2n+1$ without a contact structure.
For $n=2$, $SU(3)/SO(3)$ has no contact structure and for $n>2$,
$SU(3)/SO(3)timesmathbb{S}^{2n-4}$ has no contact structure, see Proposition 2.4 in [2].
[1] J. Martinet,
Formes de contact sur les variétés de dimension 3. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp. 142–163. Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971.
[2] R. E. Stong, Contact manifolds. J. Differential Geometry 9 (1974), 219–238.
answered 2 hours ago
Piotr HajlaszPiotr Hajlasz
9,46843672
$endgroup$
Every compact orientable $3$-dimensional manifold has a contact structure [1]. On the other hand we have
Theorem. For $ngeq 2$ there is a closed oriented connected manifold of dimension $2n+1$ without a contact structure.
For $n=2$, $SU(3)/SO(3)$ has no contact structure and for $n>2$,
$SU(3)/SO(3)timesmathbb{S}^{2n-4}$ has no contact structure, see Proposition 2.4 in [2].
[1] J. Martinet,
Formes de contact sur les variétés de dimension 3. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp. 142–163. Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971.
[2] R. E. Stong, Contact manifolds. J. Differential Geometry 9 (1974), 219–238.
answered 2 hours ago
Piotr HajlaszPiotr Hajlasz
9,46843672
answered 2 hours ago
Piotr HajlaszPiotr Hajlasz
9,46843672
answered 2 hours ago
Piotr HajlaszPiotr Hajlasz
9,46843672
answered 2 hours ago
Piotr HajlaszPiotr Hajlasz
9,46843672
9,46843672
add a comment |
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$begingroup$
Not exactly answering the question since I'm not providing examples, but for a manifold $M^{2n+1}$, it turns out that admitting a contact structure is equivalent to admitting a reduction of structure group to $U(n) times 1$ (such a reduction is called an almost contact structure), as proved by Borman, Eliashberg, and Murphy. The contact structures they produce for a given almost contact class are overtwisted, meaning they contain some model overtwisted chart. It is a more difficult question to ask when a manifold admits a tight (= non-overtwisted) contact structure.
$endgroup$
– KSackel
2 hours ago