Is a manifold-with-boundary with given interior and non-empty boundary essentially unique?Contractible manifold with boundary - is it a disc?Manifolds with two coordinate chartsDoes a *topological* manifold have an exhaustion by compact submanifolds with boundary?Exotic line arrangementsVolume form on a hyperbolic manifold with geodesic boundaryOn compact, orientable 3-manifolds with non-empty boundaryExtension of a group action beyond the boundaryRemove a disc from a manifold. When is the resulting sphere nullhomotopic?

Is a manifold-with-boundary with given interior and non-empty boundary essentially unique?


Contractible manifold with boundary - is it a disc?Manifolds with two coordinate chartsDoes a *topological* manifold have an exhaustion by compact submanifolds with boundary?Exotic line arrangementsVolume form on a hyperbolic manifold with geodesic boundaryOn compact, orientable 3-manifolds with non-empty boundaryExtension of a group action beyond the boundaryRemove a disc from a manifold. When is the resulting sphere nullhomotopic?













4












$begingroup$


Let $M$ be a compact connected manifold-with-boundary such that $circ M neq emptyset$, where $circ M$ is the boundary of $M$. Let $N$ be a compact connected manifold-with-boundary such that $circ N neq emptyset$ and $bullet M approx bullet N$, where $bullet M$ denotes the interior of $M$ and $approx$ denotes homeomorphic. Does it necessarily hold that $N approx M$?



(I have asked this question before here, but there were no replies.)










share|cite|improve this question











$endgroup$


















    4












    $begingroup$


    Let $M$ be a compact connected manifold-with-boundary such that $circ M neq emptyset$, where $circ M$ is the boundary of $M$. Let $N$ be a compact connected manifold-with-boundary such that $circ N neq emptyset$ and $bullet M approx bullet N$, where $bullet M$ denotes the interior of $M$ and $approx$ denotes homeomorphic. Does it necessarily hold that $N approx M$?



    (I have asked this question before here, but there were no replies.)










    share|cite|improve this question











    $endgroup$
















      4












      4








      4


      1



      $begingroup$


      Let $M$ be a compact connected manifold-with-boundary such that $circ M neq emptyset$, where $circ M$ is the boundary of $M$. Let $N$ be a compact connected manifold-with-boundary such that $circ N neq emptyset$ and $bullet M approx bullet N$, where $bullet M$ denotes the interior of $M$ and $approx$ denotes homeomorphic. Does it necessarily hold that $N approx M$?



      (I have asked this question before here, but there were no replies.)










      share|cite|improve this question











      $endgroup$




      Let $M$ be a compact connected manifold-with-boundary such that $circ M neq emptyset$, where $circ M$ is the boundary of $M$. Let $N$ be a compact connected manifold-with-boundary such that $circ N neq emptyset$ and $bullet M approx bullet N$, where $bullet M$ denotes the interior of $M$ and $approx$ denotes homeomorphic. Does it necessarily hold that $N approx M$?



      (I have asked this question before here, but there were no replies.)







      differential-topology manifolds






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Apr 15 at 22:33









      YCor

      31.2k4 gold badges96 silver badges148 bronze badges




      31.2k4 gold badges96 silver badges148 bronze badges










      asked Apr 15 at 20:21









      kabakaba

      1233 bronze badges




      1233 bronze badges























          1 Answer
          1






          active

          oldest

          votes


















          12














          $begingroup$

          No, there are examples detected by Whitehead torsion. If $P$ is a compact connected $(n-1)$-manifold with empty boundary, then (assuming $nge 6$) for every element $tau$ of the Whitehead group of $pi_1(P)$ there is an $h$-cobordism $M$ on $P$ such that $tau$ is the Whitehead torsion of the pair $(M,P)$. The interior of $M$ will be isomorphic to $Ptimesmathbb R$, but if $tau$ is nontrivial then $M$ will not be isomorphic to $Ptimes I$.






          share|cite|improve this answer











          $endgroup$














          • $begingroup$
            Thank you. I do not yet understand the answer; I will have to read about Whitehead torsion tomorrow. Would the same answer apply if $M$ and $N$ were orientable?
            $endgroup$
            – kaba
            Apr 15 at 23:45










          • $begingroup$
            Yes, it has nothing to do with orientability.
            $endgroup$
            – Tom Goodwillie
            Apr 16 at 0:30










          • $begingroup$
            How do I find a compact connected manifold $P$ with a non-trivial Whitehead group $Wh(pi_1(P))$?
            $endgroup$
            – kaba
            Apr 24 at 20:22










          • $begingroup$
            You can use a lens space. Many finite cyclic groups have nontrivial Whitehead groups.
            $endgroup$
            – Tom Goodwillie
            Apr 24 at 22:35













          Your Answer








          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/4.0/"u003ecc by-sa 4.0 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%2f328149%2fis-a-manifold-with-boundary-with-given-interior-and-non-empty-boundary-essential%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









          12














          $begingroup$

          No, there are examples detected by Whitehead torsion. If $P$ is a compact connected $(n-1)$-manifold with empty boundary, then (assuming $nge 6$) for every element $tau$ of the Whitehead group of $pi_1(P)$ there is an $h$-cobordism $M$ on $P$ such that $tau$ is the Whitehead torsion of the pair $(M,P)$. The interior of $M$ will be isomorphic to $Ptimesmathbb R$, but if $tau$ is nontrivial then $M$ will not be isomorphic to $Ptimes I$.






          share|cite|improve this answer











          $endgroup$














          • $begingroup$
            Thank you. I do not yet understand the answer; I will have to read about Whitehead torsion tomorrow. Would the same answer apply if $M$ and $N$ were orientable?
            $endgroup$
            – kaba
            Apr 15 at 23:45










          • $begingroup$
            Yes, it has nothing to do with orientability.
            $endgroup$
            – Tom Goodwillie
            Apr 16 at 0:30










          • $begingroup$
            How do I find a compact connected manifold $P$ with a non-trivial Whitehead group $Wh(pi_1(P))$?
            $endgroup$
            – kaba
            Apr 24 at 20:22










          • $begingroup$
            You can use a lens space. Many finite cyclic groups have nontrivial Whitehead groups.
            $endgroup$
            – Tom Goodwillie
            Apr 24 at 22:35















          12














          $begingroup$

          No, there are examples detected by Whitehead torsion. If $P$ is a compact connected $(n-1)$-manifold with empty boundary, then (assuming $nge 6$) for every element $tau$ of the Whitehead group of $pi_1(P)$ there is an $h$-cobordism $M$ on $P$ such that $tau$ is the Whitehead torsion of the pair $(M,P)$. The interior of $M$ will be isomorphic to $Ptimesmathbb R$, but if $tau$ is nontrivial then $M$ will not be isomorphic to $Ptimes I$.






          share|cite|improve this answer











          $endgroup$














          • $begingroup$
            Thank you. I do not yet understand the answer; I will have to read about Whitehead torsion tomorrow. Would the same answer apply if $M$ and $N$ were orientable?
            $endgroup$
            – kaba
            Apr 15 at 23:45










          • $begingroup$
            Yes, it has nothing to do with orientability.
            $endgroup$
            – Tom Goodwillie
            Apr 16 at 0:30










          • $begingroup$
            How do I find a compact connected manifold $P$ with a non-trivial Whitehead group $Wh(pi_1(P))$?
            $endgroup$
            – kaba
            Apr 24 at 20:22










          • $begingroup$
            You can use a lens space. Many finite cyclic groups have nontrivial Whitehead groups.
            $endgroup$
            – Tom Goodwillie
            Apr 24 at 22:35













          12














          12










          12







          $begingroup$

          No, there are examples detected by Whitehead torsion. If $P$ is a compact connected $(n-1)$-manifold with empty boundary, then (assuming $nge 6$) for every element $tau$ of the Whitehead group of $pi_1(P)$ there is an $h$-cobordism $M$ on $P$ such that $tau$ is the Whitehead torsion of the pair $(M,P)$. The interior of $M$ will be isomorphic to $Ptimesmathbb R$, but if $tau$ is nontrivial then $M$ will not be isomorphic to $Ptimes I$.






          share|cite|improve this answer











          $endgroup$



          No, there are examples detected by Whitehead torsion. If $P$ is a compact connected $(n-1)$-manifold with empty boundary, then (assuming $nge 6$) for every element $tau$ of the Whitehead group of $pi_1(P)$ there is an $h$-cobordism $M$ on $P$ such that $tau$ is the Whitehead torsion of the pair $(M,P)$. The interior of $M$ will be isomorphic to $Ptimesmathbb R$, but if $tau$ is nontrivial then $M$ will not be isomorphic to $Ptimes I$.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Apr 15 at 22:14

























          answered Apr 15 at 21:15









          Tom GoodwillieTom Goodwillie

          41.4k3 gold badges113 silver badges203 bronze badges




          41.4k3 gold badges113 silver badges203 bronze badges














          • $begingroup$
            Thank you. I do not yet understand the answer; I will have to read about Whitehead torsion tomorrow. Would the same answer apply if $M$ and $N$ were orientable?
            $endgroup$
            – kaba
            Apr 15 at 23:45










          • $begingroup$
            Yes, it has nothing to do with orientability.
            $endgroup$
            – Tom Goodwillie
            Apr 16 at 0:30










          • $begingroup$
            How do I find a compact connected manifold $P$ with a non-trivial Whitehead group $Wh(pi_1(P))$?
            $endgroup$
            – kaba
            Apr 24 at 20:22










          • $begingroup$
            You can use a lens space. Many finite cyclic groups have nontrivial Whitehead groups.
            $endgroup$
            – Tom Goodwillie
            Apr 24 at 22:35
















          • $begingroup$
            Thank you. I do not yet understand the answer; I will have to read about Whitehead torsion tomorrow. Would the same answer apply if $M$ and $N$ were orientable?
            $endgroup$
            – kaba
            Apr 15 at 23:45










          • $begingroup$
            Yes, it has nothing to do with orientability.
            $endgroup$
            – Tom Goodwillie
            Apr 16 at 0:30










          • $begingroup$
            How do I find a compact connected manifold $P$ with a non-trivial Whitehead group $Wh(pi_1(P))$?
            $endgroup$
            – kaba
            Apr 24 at 20:22










          • $begingroup$
            You can use a lens space. Many finite cyclic groups have nontrivial Whitehead groups.
            $endgroup$
            – Tom Goodwillie
            Apr 24 at 22:35















          $begingroup$
          Thank you. I do not yet understand the answer; I will have to read about Whitehead torsion tomorrow. Would the same answer apply if $M$ and $N$ were orientable?
          $endgroup$
          – kaba
          Apr 15 at 23:45




          $begingroup$
          Thank you. I do not yet understand the answer; I will have to read about Whitehead torsion tomorrow. Would the same answer apply if $M$ and $N$ were orientable?
          $endgroup$
          – kaba
          Apr 15 at 23:45












          $begingroup$
          Yes, it has nothing to do with orientability.
          $endgroup$
          – Tom Goodwillie
          Apr 16 at 0:30




          $begingroup$
          Yes, it has nothing to do with orientability.
          $endgroup$
          – Tom Goodwillie
          Apr 16 at 0:30












          $begingroup$
          How do I find a compact connected manifold $P$ with a non-trivial Whitehead group $Wh(pi_1(P))$?
          $endgroup$
          – kaba
          Apr 24 at 20:22




          $begingroup$
          How do I find a compact connected manifold $P$ with a non-trivial Whitehead group $Wh(pi_1(P))$?
          $endgroup$
          – kaba
          Apr 24 at 20:22












          $begingroup$
          You can use a lens space. Many finite cyclic groups have nontrivial Whitehead groups.
          $endgroup$
          – Tom Goodwillie
          Apr 24 at 22:35




          $begingroup$
          You can use a lens space. Many finite cyclic groups have nontrivial Whitehead groups.
          $endgroup$
          – Tom Goodwillie
          Apr 24 at 22:35


















          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%2f328149%2fis-a-manifold-with-boundary-with-given-interior-and-non-empty-boundary-essential%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

          Tamil (spriik) Luke uk diar | Nawigatjuun

          Align equal signs while including text over equalitiesAMS align: left aligned text/math plus multicolumn alignmentMultiple alignmentsAligning equations in multiple placesNumbering and aligning an equation with multiple columnsHow to align one equation with another multline equationUsing \ in environments inside the begintabularxNumber equations and preserving alignment of equal signsHow can I align equations to the left and to the right?Double equation alignment problem within align enviromentAligned within align: Why are they right-aligned?

          Training a classifier when some of the features are unknownWhy does Gradient Boosting regression predict negative values when there are no negative y-values in my training set?How to improve an existing (trained) classifier?What is effect when I set up some self defined predisctor variables?Why Matlab neural network classification returns decimal values on prediction dataset?Fitting and transforming text data in training, testing, and validation setsHow to quantify the performance of the classifier (multi-class SVM) using the test data?How do I control for some patients providing multiple samples in my training data?Training and Test setTraining a convolutional neural network for image denoising in MatlabShouldn't an autoencoder with #(neurons in hidden layer) = #(neurons in input layer) be “perfect”?