What is the precise definition of a multi-connected manifold?












1












$begingroup$


I am looking for a term to describe manifolds that are connected but not simply connected. Multi-connected looks like a strong candidate. However, I can't seem to find a formal definition of the concept. What is the precise definition of a multi-connected manifold?






general-topology terminology







share|cite|improve this question









$endgroup$








  • 2




    $begingroup$
    You already gave it: a manifold that's connected but not simply connected. I think this is old language, though.
    $endgroup$
    – Qiaochu Yuan
    Dec 17 '18 at 0:07










  • $begingroup$
    How would you term it?
    $endgroup$
    – Asdf
    Dec 17 '18 at 0:14






  • 2




    $begingroup$
    Just "connected but not simply connected." It's not a concept I have much of a need for so I'm okay not having a term for it.
    $endgroup$
    – Qiaochu Yuan
    Dec 17 '18 at 0:15










  • $begingroup$
    Alright. I'll take multi-connected.
    $endgroup$
    – Asdf
    Dec 17 '18 at 0:18
















1












$begingroup$


I am looking for a term to describe manifolds that are connected but not simply connected. Multi-connected looks like a strong candidate. However, I can't seem to find a formal definition of the concept. What is the precise definition of a multi-connected manifold?






general-topology terminology







share|cite|improve this question









$endgroup$








  • 2




    $begingroup$
    You already gave it: a manifold that's connected but not simply connected. I think this is old language, though.
    $endgroup$
    – Qiaochu Yuan
    Dec 17 '18 at 0:07










  • $begingroup$
    How would you term it?
    $endgroup$
    – Asdf
    Dec 17 '18 at 0:14






  • 2




    $begingroup$
    Just "connected but not simply connected." It's not a concept I have much of a need for so I'm okay not having a term for it.
    $endgroup$
    – Qiaochu Yuan
    Dec 17 '18 at 0:15










  • $begingroup$
    Alright. I'll take multi-connected.
    $endgroup$
    – Asdf
    Dec 17 '18 at 0:18














1












1








1





$begingroup$


I am looking for a term to describe manifolds that are connected but not simply connected. Multi-connected looks like a strong candidate. However, I can't seem to find a formal definition of the concept. What is the precise definition of a multi-connected manifold?






general-topology terminology







share|cite|improve this question









$endgroup$




I am looking for a term to describe manifolds that are connected but not simply connected. Multi-connected looks like a strong candidate. However, I can't seem to find a formal definition of the concept. What is the precise definition of a multi-connected manifold?






general-topology terminology




general-topology terminology






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 17 '18 at 0:01









AsdfAsdf

417213




417213








  • 2




    $begingroup$
    You already gave it: a manifold that's connected but not simply connected. I think this is old language, though.
    $endgroup$
    – Qiaochu Yuan
    Dec 17 '18 at 0:07










  • $begingroup$
    How would you term it?
    $endgroup$
    – Asdf
    Dec 17 '18 at 0:14






  • 2




    $begingroup$
    Just "connected but not simply connected." It's not a concept I have much of a need for so I'm okay not having a term for it.
    $endgroup$
    – Qiaochu Yuan
    Dec 17 '18 at 0:15










  • $begingroup$
    Alright. I'll take multi-connected.
    $endgroup$
    – Asdf
    Dec 17 '18 at 0:18














  • 2




    $begingroup$
    You already gave it: a manifold that's connected but not simply connected. I think this is old language, though.
    $endgroup$
    – Qiaochu Yuan
    Dec 17 '18 at 0:07










  • $begingroup$
    How would you term it?
    $endgroup$
    – Asdf
    Dec 17 '18 at 0:14






  • 2




    $begingroup$
    Just "connected but not simply connected." It's not a concept I have much of a need for so I'm okay not having a term for it.
    $endgroup$
    – Qiaochu Yuan
    Dec 17 '18 at 0:15










  • $begingroup$
    Alright. I'll take multi-connected.
    $endgroup$
    – Asdf
    Dec 17 '18 at 0:18








2




2




$begingroup$
You already gave it: a manifold that's connected but not simply connected. I think this is old language, though.
$endgroup$
– Qiaochu Yuan
Dec 17 '18 at 0:07




$begingroup$
You already gave it: a manifold that's connected but not simply connected. I think this is old language, though.
$endgroup$
– Qiaochu Yuan
Dec 17 '18 at 0:07












$begingroup$
How would you term it?
$endgroup$
– Asdf
Dec 17 '18 at 0:14




$begingroup$
How would you term it?
$endgroup$
– Asdf
Dec 17 '18 at 0:14




2




2




$begingroup$
Just "connected but not simply connected." It's not a concept I have much of a need for so I'm okay not having a term for it.
$endgroup$
– Qiaochu Yuan
Dec 17 '18 at 0:15




$begingroup$
Just "connected but not simply connected." It's not a concept I have much of a need for so I'm okay not having a term for it.
$endgroup$
– Qiaochu Yuan
Dec 17 '18 at 0:15












$begingroup$
Alright. I'll take multi-connected.
$endgroup$
– Asdf
Dec 17 '18 at 0:18




$begingroup$
Alright. I'll take multi-connected.
$endgroup$
– Asdf
Dec 17 '18 at 0:18










2 Answers
2






active

oldest

votes


















1












$begingroup$

I am not convinced that the expression "multi-connected" is a good choice for a connected but non-simply connected manifold.



The expression "doubly connected" is frequently used in complex analysis. A doubly connected region in the complex plane is a region bounded by two Jordan curves. See for example Conformal map of doubly connected domain into annulus..



A generalization is "multiply connected". See https://www.encyclopediaofmath.org/index.php/Multiply-connected_domain. Here are some references.



Walsh, J. L. "On the conformal mapping of multiply connected regions." Transactions of the American Mathematical Society 82.1 (1956): 128-146.



https://www.ams.org/journals/tran/1956-082-01/S0002-9947-1956-0080727-2/S0002-9947-1956-0080727-2.pdf



Walsh, J. L., and H. J. Landau. "On canonical conformal maps of multiply connected regions." Transactions of the American Mathematical Society 93.1 (1959): 81-96.



https://www.ams.org/journals/tran/1959-093-01/S0002-9947-1959-0160884-2/S0002-9947-1959-0160884-2.pdf



Landau, H. J. "On canonical conformal maps of multiply connected domains." Transactions of the American Mathematical Society 99.1 (1961): 1-20.



https://www.ams.org/journals/tran/1961-099-01/S0002-9947-1961-0121474-X/S0002-9947-1961-0121474-X.pdf






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Multiply connected looks like well-established terminology. It has a Mathworld page mathworld.wolfram.com/MultiplyConnected.html I found multi-connected in some physics literature, but it bothered me that it did not seem to turn up in the mathematics literature.
    $endgroup$
    – Asdf
    Dec 18 '18 at 5:33





















1












$begingroup$

A manifold that is connected (so path-connected too) but not simply connected, i.e. $pi_1(X,x_0)$ is not trivial (for some choice of base point, it matters not which, by path-connectedness).






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: "69"
    };
    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%2fmath.stackexchange.com%2fquestions%2f3043366%2fwhat-is-the-precise-definition-of-a-multi-connected-manifold%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1












    $begingroup$

    I am not convinced that the expression "multi-connected" is a good choice for a connected but non-simply connected manifold.



    The expression "doubly connected" is frequently used in complex analysis. A doubly connected region in the complex plane is a region bounded by two Jordan curves. See for example Conformal map of doubly connected domain into annulus..



    A generalization is "multiply connected". See https://www.encyclopediaofmath.org/index.php/Multiply-connected_domain. Here are some references.



    Walsh, J. L. "On the conformal mapping of multiply connected regions." Transactions of the American Mathematical Society 82.1 (1956): 128-146.



    https://www.ams.org/journals/tran/1956-082-01/S0002-9947-1956-0080727-2/S0002-9947-1956-0080727-2.pdf



    Walsh, J. L., and H. J. Landau. "On canonical conformal maps of multiply connected regions." Transactions of the American Mathematical Society 93.1 (1959): 81-96.



    https://www.ams.org/journals/tran/1959-093-01/S0002-9947-1959-0160884-2/S0002-9947-1959-0160884-2.pdf



    Landau, H. J. "On canonical conformal maps of multiply connected domains." Transactions of the American Mathematical Society 99.1 (1961): 1-20.



    https://www.ams.org/journals/tran/1961-099-01/S0002-9947-1961-0121474-X/S0002-9947-1961-0121474-X.pdf






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Multiply connected looks like well-established terminology. It has a Mathworld page mathworld.wolfram.com/MultiplyConnected.html I found multi-connected in some physics literature, but it bothered me that it did not seem to turn up in the mathematics literature.
      $endgroup$
      – Asdf
      Dec 18 '18 at 5:33


















    1












    $begingroup$

    I am not convinced that the expression "multi-connected" is a good choice for a connected but non-simply connected manifold.



    The expression "doubly connected" is frequently used in complex analysis. A doubly connected region in the complex plane is a region bounded by two Jordan curves. See for example Conformal map of doubly connected domain into annulus..



    A generalization is "multiply connected". See https://www.encyclopediaofmath.org/index.php/Multiply-connected_domain. Here are some references.



    Walsh, J. L. "On the conformal mapping of multiply connected regions." Transactions of the American Mathematical Society 82.1 (1956): 128-146.



    https://www.ams.org/journals/tran/1956-082-01/S0002-9947-1956-0080727-2/S0002-9947-1956-0080727-2.pdf



    Walsh, J. L., and H. J. Landau. "On canonical conformal maps of multiply connected regions." Transactions of the American Mathematical Society 93.1 (1959): 81-96.



    https://www.ams.org/journals/tran/1959-093-01/S0002-9947-1959-0160884-2/S0002-9947-1959-0160884-2.pdf



    Landau, H. J. "On canonical conformal maps of multiply connected domains." Transactions of the American Mathematical Society 99.1 (1961): 1-20.



    https://www.ams.org/journals/tran/1961-099-01/S0002-9947-1961-0121474-X/S0002-9947-1961-0121474-X.pdf






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Multiply connected looks like well-established terminology. It has a Mathworld page mathworld.wolfram.com/MultiplyConnected.html I found multi-connected in some physics literature, but it bothered me that it did not seem to turn up in the mathematics literature.
      $endgroup$
      – Asdf
      Dec 18 '18 at 5:33
















    1












    1








    1





    $begingroup$

    I am not convinced that the expression "multi-connected" is a good choice for a connected but non-simply connected manifold.



    The expression "doubly connected" is frequently used in complex analysis. A doubly connected region in the complex plane is a region bounded by two Jordan curves. See for example Conformal map of doubly connected domain into annulus..



    A generalization is "multiply connected". See https://www.encyclopediaofmath.org/index.php/Multiply-connected_domain. Here are some references.



    Walsh, J. L. "On the conformal mapping of multiply connected regions." Transactions of the American Mathematical Society 82.1 (1956): 128-146.



    https://www.ams.org/journals/tran/1956-082-01/S0002-9947-1956-0080727-2/S0002-9947-1956-0080727-2.pdf



    Walsh, J. L., and H. J. Landau. "On canonical conformal maps of multiply connected regions." Transactions of the American Mathematical Society 93.1 (1959): 81-96.



    https://www.ams.org/journals/tran/1959-093-01/S0002-9947-1959-0160884-2/S0002-9947-1959-0160884-2.pdf



    Landau, H. J. "On canonical conformal maps of multiply connected domains." Transactions of the American Mathematical Society 99.1 (1961): 1-20.



    https://www.ams.org/journals/tran/1961-099-01/S0002-9947-1961-0121474-X/S0002-9947-1961-0121474-X.pdf






    share|cite|improve this answer









    $endgroup$



    I am not convinced that the expression "multi-connected" is a good choice for a connected but non-simply connected manifold.



    The expression "doubly connected" is frequently used in complex analysis. A doubly connected region in the complex plane is a region bounded by two Jordan curves. See for example Conformal map of doubly connected domain into annulus..



    A generalization is "multiply connected". See https://www.encyclopediaofmath.org/index.php/Multiply-connected_domain. Here are some references.



    Walsh, J. L. "On the conformal mapping of multiply connected regions." Transactions of the American Mathematical Society 82.1 (1956): 128-146.



    https://www.ams.org/journals/tran/1956-082-01/S0002-9947-1956-0080727-2/S0002-9947-1956-0080727-2.pdf



    Walsh, J. L., and H. J. Landau. "On canonical conformal maps of multiply connected regions." Transactions of the American Mathematical Society 93.1 (1959): 81-96.



    https://www.ams.org/journals/tran/1959-093-01/S0002-9947-1959-0160884-2/S0002-9947-1959-0160884-2.pdf



    Landau, H. J. "On canonical conformal maps of multiply connected domains." Transactions of the American Mathematical Society 99.1 (1961): 1-20.



    https://www.ams.org/journals/tran/1961-099-01/S0002-9947-1961-0121474-X/S0002-9947-1961-0121474-X.pdf







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Dec 17 '18 at 14:29









    Paul FrostPaul Frost

    10.7k3933




    10.7k3933












    • $begingroup$
      Multiply connected looks like well-established terminology. It has a Mathworld page mathworld.wolfram.com/MultiplyConnected.html I found multi-connected in some physics literature, but it bothered me that it did not seem to turn up in the mathematics literature.
      $endgroup$
      – Asdf
      Dec 18 '18 at 5:33




















    • $begingroup$
      Multiply connected looks like well-established terminology. It has a Mathworld page mathworld.wolfram.com/MultiplyConnected.html I found multi-connected in some physics literature, but it bothered me that it did not seem to turn up in the mathematics literature.
      $endgroup$
      – Asdf
      Dec 18 '18 at 5:33


















    $begingroup$
    Multiply connected looks like well-established terminology. It has a Mathworld page mathworld.wolfram.com/MultiplyConnected.html I found multi-connected in some physics literature, but it bothered me that it did not seem to turn up in the mathematics literature.
    $endgroup$
    – Asdf
    Dec 18 '18 at 5:33






    $begingroup$
    Multiply connected looks like well-established terminology. It has a Mathworld page mathworld.wolfram.com/MultiplyConnected.html I found multi-connected in some physics literature, but it bothered me that it did not seem to turn up in the mathematics literature.
    $endgroup$
    – Asdf
    Dec 18 '18 at 5:33













    1












    $begingroup$

    A manifold that is connected (so path-connected too) but not simply connected, i.e. $pi_1(X,x_0)$ is not trivial (for some choice of base point, it matters not which, by path-connectedness).






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      A manifold that is connected (so path-connected too) but not simply connected, i.e. $pi_1(X,x_0)$ is not trivial (for some choice of base point, it matters not which, by path-connectedness).






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        A manifold that is connected (so path-connected too) but not simply connected, i.e. $pi_1(X,x_0)$ is not trivial (for some choice of base point, it matters not which, by path-connectedness).






        share|cite|improve this answer









        $endgroup$



        A manifold that is connected (so path-connected too) but not simply connected, i.e. $pi_1(X,x_0)$ is not trivial (for some choice of base point, it matters not which, by path-connectedness).







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 17 '18 at 10:56









        Henno BrandsmaHenno Brandsma

        109k347115




        109k347115






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • 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%2fmath.stackexchange.com%2fquestions%2f3043366%2fwhat-is-the-precise-definition-of-a-multi-connected-manifold%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

            Ellipse (mathématiques)

            Image
            Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti...
            Read more

            Quarter-circle Tiles

            Image
            up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2 ...
            Read more

            Mont Emei

            Image
            Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
            Read more