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












4












$begingroup$


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
    3 hours ago
















4












$begingroup$


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
    3 hours ago














4












4








4


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













share|cite|improve this question




share|cite|improve this question








edited 2 hours ago









Piotr Hajlasz

9,46843672




9,46843672










asked 3 hours ago









Warlock of Firetop MountainWarlock of Firetop Mountain

25217




25217








  • 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
    3 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
    3 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
3 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
3 hours ago










1 Answer
1






active

oldest

votes


















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









$endgroup$













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "504"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f325397%2fexamples-of-odd-dimensional-manifolds-that-do-not-admit-contact-structure%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    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 3 hours ago









        Piotr HajlaszPiotr Hajlasz

        9,46843672




        9,46843672






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to MathOverflow!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f325397%2fexamples-of-odd-dimensional-manifolds-that-do-not-admit-contact-structure%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            SQL Server 17 - Attemping to backup to remote NAS but Access is denied

            Image
            .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ margin-bottom:0; } 0 I was attemping to use Ola Hallengrens scripts to backup to a remote NAS, but it was failing and not giving any direct errors, so I tried doing it manually using Server Management Studio and got the following error: TITLE: Microsoft SQL Server Management Studio Failed to retrieve data for this request. (Microsoft.SqlServer.Management.Sdk.Sfc) For help, click: http://go.microsoft.com/fwlink?ProdName=Microsoft%20SQL%20Server&LinkId=20476 ADDITIONAL INFORMATION: An exception occurred while executing a Transact-SQL statement or batch. (Microsoft.SqlServer.ConnectionInfo) The operating system returned the error '0x80070005(Access is denied.)&#
            Read more

            Always On Availability groups resolving state after failover - Remote harden of transaction...

            Image
            0 My explanation will be opinion based as it is based on my findings. If more information is needed, I am always happy to add it. The setup We have 4 SQL Server 2016 Instances, and all 4 underlying servers are in a WSFC (In my eyes a strange setup since Always On AG's are 2 by 2. But that is a different question.) 4 nodes + Fileshare witness Version: 13.0.5026.0 (all nodes) The instances that are having the failover issues are instances 1 & 2. They have 3 AG's between them. AG1 is primary on instance 1 and is having the issue. AG2 and AG3 are primary on instance 2. Instances 3 and 4 have their own separate AG's (2). AG1's settings The week before Last week, after checking for long running queries (none) and the synchronization state (green accross the board) we initiated a
            Read more

            Restoring from pg_dump with foreign key constraints

            Image
            5 2 In restoring a database from a pg_dump , a number of errors are being generated and the whole table is subsequently being ignored. An example: ERROR: insert or update on table "channelproducts" violates foreign key constraint "fk_rails_dfaae373a5" DETAIL: Key (channel_id)=(1) is not present in table "channels". Interestingly enough, I've noted that all these instances are popping up because of the sequence of loading. channels is after channelproducts both alphabetically and in the file and thus I can understand why postgres complains about having to create a child without a parent. caveat: the foreign key is being generated automatically by rails 4.2: I could remove the problem at the source but that still does not really solve the problem... version: PostgreSQL
            Read more