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?
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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.)
differential-topology manifolds
edited Apr 15 at 22:33
YCor
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asked Apr 15 at 20:21
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.)
differential-topology manifolds
edited Apr 15 at 22:33
YCor
31.2k4 gold badges96 silver badges148 bronze badges
asked Apr 15 at 20:21
kabakaba
1233 bronze badges
$endgroup$
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.)
differential-topology manifolds
edited Apr 15 at 22:33
YCor
31.2k4 gold badges96 silver badges148 bronze badges
asked Apr 15 at 20:21
kabakaba
1233 bronze badges
$endgroup$
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
differential-topology manifolds
edited Apr 15 at 22:33
YCor
31.2k4 gold badges96 silver badges148 bronze badges
asked Apr 15 at 20:21
kabakaba
1233 bronze badges
edited Apr 15 at 22:33
YCor
31.2k4 gold badges96 silver badges148 bronze badges
asked Apr 15 at 20:21
kabakaba
1233 bronze badges
edited Apr 15 at 22:33
YCor
31.2k4 gold badges96 silver badges148 bronze badges
edited Apr 15 at 22:33
YCor
31.2k4 gold badges96 silver badges148 bronze badges
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
asked Apr 15 at 20:21
kabakaba
1233 bronze badges
asked Apr 15 at 20:21
kabakaba
1233 bronze badges
1233 bronze badges
add a comment
|
add a comment
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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 $nge 6$) 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 $Ptimes I$.
answered Apr 15 at 21:15
Tom GoodwillieTom Goodwillie
41.4k3 gold badges113 silver badges203 bronze badges
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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
add a comment
|
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1 Answer
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1 Answer
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oldest
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active
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$begingroup$
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 $Ptimes I$.
answered Apr 15 at 21:15
Tom GoodwillieTom Goodwillie
41.4k3 gold badges113 silver badges203 bronze badges
$endgroup$
$begingroup$
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
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 $nge 6$) 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 $Ptimes I$.
answered Apr 15 at 21:15
Tom GoodwillieTom Goodwillie
41.4k3 gold badges113 silver badges203 bronze badges
$endgroup$
$begingroup$
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
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 $nge 6$) 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 $Ptimes I$.
answered Apr 15 at 21:15
Tom GoodwillieTom Goodwillie
41.4k3 gold badges113 silver badges203 bronze badges
$endgroup$
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 $Ptimes I$.
answered Apr 15 at 21:15
Tom GoodwillieTom Goodwillie
41.4k3 gold badges113 silver badges203 bronze badges
edited Apr 15 at 22:14
answered Apr 15 at 21:15
Tom GoodwillieTom Goodwillie
41.4k3 gold badges113 silver badges203 bronze badges
answered Apr 15 at 21:15
Tom GoodwillieTom Goodwillie
41.4k3 gold badges113 silver badges203 bronze badges
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
add a comment
|
$begingroup$
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
add a comment
|
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