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
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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
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kabakaba
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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.
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add a comment |
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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.
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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
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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.
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1 Answer
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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 36 mins ago
Tom GoodwillieTom Goodwillie
40.3k3110200
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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 36 mins ago
Tom GoodwillieTom Goodwillie
40.3k3110200
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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 36 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 36 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 36 mins ago
Tom GoodwillieTom Goodwillie
40.3k3110200
answered 36 mins ago
Tom GoodwillieTom Goodwillie
40.3k3110200
answered 36 mins ago
Tom GoodwillieTom Goodwillie
40.3k3110200
answered 36 mins ago
Tom GoodwillieTom Goodwillie
40.3k3110200
40.3k3110200
add a comment |
add a comment |
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