Is a manifold-with-boundary with given interior and non-empty boundary essentially unique? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)How does the Lefschetz-Poincare dual torsion linking pairing on manifolds with boundary interact with the maps of the long exact sequence of the manifold-boundary pair?fundamental domain of universal coveringhomotopy type of embeddings versus diffeomorphismsDoes a *topological* manifold have an exhaustion by compact submanifolds with boundary?Is the complement of the ends of a manifold bounded?On compact, orientable 3-manifolds with non-empty boundaryManifolds from fundamental piecesRemove a disc from a manifold. When is the resulting sphere nullhomotopic?Can we cut and rotate a particular region of a hyperbolic 3-manifold to get another (non-homeomorphic) hyperbolic 3-manifold?Irreducible separators of compact manifolds
Is a manifold-with-boundary with given interior and non-empty boundary essentially unique?
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)How does the Lefschetz-Poincare dual torsion linking pairing on manifolds with boundary interact with the maps of the long exact sequence of the manifold-boundary pair?fundamental domain of universal coveringhomotopy type of embeddings versus diffeomorphismsDoes a *topological* manifold have an exhaustion by compact submanifolds with boundary?Is the complement of the ends of a manifold bounded?On compact, orientable 3-manifolds with non-empty boundaryManifolds from fundamental piecesRemove a disc from a manifold. When is the resulting sphere nullhomotopic?Can we cut and rotate a particular region of a hyperbolic 3-manifold to get another (non-homeomorphic) hyperbolic 3-manifold?Irreducible separators of compact manifolds
$begingroup$
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.)
manifolds
asked 1 hour ago
kabakaba
1111
New contributor
kaba is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
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.)
manifolds
asked 1 hour ago
kabakaba
1111
New contributor
kaba is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
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.)
manifolds
asked 1 hour ago
kabakaba
1111
New contributor
kaba is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
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.)
manifolds
manifolds
asked 1 hour ago
kabakaba
1111
New contributor
kaba is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
kabakaba
1111
New contributor
kaba is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
kabakaba
1111
New contributor
kaba is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
kabakaba
1111
asked 1 hour ago
kabakaba
1111
1111
New contributor
kaba is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
kaba is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
kaba is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
No, there are examples detected by Whitehead torsion. If $P$ is a compact connected $(n-1)$-manifold with empty boundary, then (assuming $nge 5$) 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 $Ptimes I$.
answered 46 mins ago
Tom GoodwillieTom Goodwillie
40.3k3110200
$endgroup$
add a comment |
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/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
);
);
kaba is a new contributor. Be nice, and check out our Code of Conduct.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
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
$begingroup$
No, there are examples detected by Whitehead torsion. If $P$ is a compact connected $(n-1)$-manifold with empty boundary, then (assuming $nge 5$) 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 $Ptimes I$.
answered 46 mins ago
Tom GoodwillieTom Goodwillie
40.3k3110200
$endgroup$
add a comment |
$begingroup$
No, there are examples detected by Whitehead torsion. If $P$ is a compact connected $(n-1)$-manifold with empty boundary, then (assuming $nge 5$) 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 $Ptimes I$.
answered 46 mins ago
Tom GoodwillieTom Goodwillie
40.3k3110200
$endgroup$
add a comment |
$begingroup$
No, there are examples detected by Whitehead torsion. If $P$ is a compact connected $(n-1)$-manifold with empty boundary, then (assuming $nge 5$) 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 $Ptimes I$.
answered 46 mins ago
Tom GoodwillieTom Goodwillie
40.3k3110200
$endgroup$
No, there are examples detected by Whitehead torsion. If $P$ is a compact connected $(n-1)$-manifold with empty boundary, then (assuming $nge 5$) 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 $Ptimes I$.
answered 46 mins ago
Tom GoodwillieTom Goodwillie
40.3k3110200
answered 46 mins ago
Tom GoodwillieTom Goodwillie
40.3k3110200
answered 46 mins ago
Tom GoodwillieTom Goodwillie
40.3k3110200
answered 46 mins ago
Tom GoodwillieTom Goodwillie
40.3k3110200
40.3k3110200
add a comment |
add a comment |
kaba is a new contributor. Be nice, and check out our Code of Conduct.
kaba is a new contributor. Be nice, and check out our Code of Conduct.
kaba is a new contributor. Be nice, and check out our Code of Conduct.
kaba is a new contributor. Be nice, and check out our Code of Conduct.
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.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
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
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
