Examples of odd-dimensional manifolds that do not admit contact structurethe existence of (almost) contact...



Examples of odd-dimensional manifolds that do not admit contact structure


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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







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  • 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
















3












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 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














3












3








3


2



$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













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edited 1 hour ago









Piotr Hajlasz

9,46843672




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asked 2 hours ago









Warlock of Firetop MountainWarlock of Firetop Mountain

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  • 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




    $begingroup$
    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










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.






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    1 Answer
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    1 Answer
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    active

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    active

    oldest

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    active

    oldest

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    5












    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$


















      5












      $begingroup$

      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.






      share|cite|improve this answer









      $endgroup$
















        5












        5








        5





        $begingroup$

        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.






        share|cite|improve this answer









        $endgroup$



        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.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 2 hours ago









        Piotr HajlaszPiotr Hajlasz

        9,46843672




        9,46843672






























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