$4D$ analog of Alexander's Horned Sphere
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Is it a trivial exercise to find a $4D$ analog to Alexander's Horned Sphere? In other words, is there a manifold homeomorphic to $S^3$, embedded in $mathbb R^4$ that has a wild complement?
Thank you.
general-topology
edited Sep 2 at 7:56
Jneven
633320
asked Sep 2 at 7:43
Barnaby Finch
111
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Is it a trivial exercise to find a $4D$ analog to Alexander's Horned Sphere? In other words, is there a manifold homeomorphic to $S^3$, embedded in $mathbb R^4$ that has a wild complement?
Thank you.
general-topology
edited Sep 2 at 7:56
Jneven
633320
asked Sep 2 at 7:43
Barnaby Finch
111
I don't think anything concerning Alexander's horned sphere is trivial...
– Lord Shark the Unknown
Sep 2 at 9:39
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Is it a trivial exercise to find a $4D$ analog to Alexander's Horned Sphere? In other words, is there a manifold homeomorphic to $S^3$, embedded in $mathbb R^4$ that has a wild complement?
Thank you.
general-topology
edited Sep 2 at 7:56
Jneven
633320
asked Sep 2 at 7:43
Barnaby Finch
111
Is it a trivial exercise to find a $4D$ analog to Alexander's Horned Sphere? In other words, is there a manifold homeomorphic to $S^3$, embedded in $mathbb R^4$ that has a wild complement?
Thank you.
general-topology
general-topology
edited Sep 2 at 7:56
Jneven
633320
asked Sep 2 at 7:43
Barnaby Finch
111
edited Sep 2 at 7:56
Jneven
633320
asked Sep 2 at 7:43
Barnaby Finch
111
edited Sep 2 at 7:56
Jneven
633320
edited Sep 2 at 7:56
Jneven
633320
edited Sep 2 at 7:56
Jneven
633320
633320
asked Sep 2 at 7:43
Barnaby Finch
111
asked Sep 2 at 7:43
Barnaby Finch
111
asked Sep 2 at 7:43
Barnaby Finch
111
111
I don't think anything concerning Alexander's horned sphere is trivial...
– Lord Shark the Unknown
Sep 2 at 9:39
add a comment |Â
I don't think anything concerning Alexander's horned sphere is trivial...
– Lord Shark the Unknown
Sep 2 at 9:39
I don't think anything concerning Alexander's horned sphere is trivial...
– Lord Shark the Unknown
Sep 2 at 9:39
I don't think anything concerning Alexander's horned sphere is trivial...
– Lord Shark the Unknown
Sep 2 at 9:39
add a comment |Â
1 Answer
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I recommend to have a look into
Rushing, T. Benny. Topological embeddings. Vol. 52. Academic Press, 1973.
Theorem 2.6.1 gives a positive answer to your question. But this is not a "trivial exercise".
However, if you have wild sphere $S subset mathbbR^n$, then you get an obvious embedding of the suspension $Sigma S$ into $mathbbR^n+1$.
If $mathbbR^n backslash S$ is not uniformly locally 1-connected (1-ULC, see Rushing), then one can show that $mathbbR^n+1 backslash Sigma S$ also fails to be 1-ULC. Thus $Sigma S subset mathbbR^n+1$ is again a wild sphere. This applies to the horned sphere.
answered Sep 2 at 13:05
Paul Frost
4,838424
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
I recommend to have a look into
Rushing, T. Benny. Topological embeddings. Vol. 52. Academic Press, 1973.
Theorem 2.6.1 gives a positive answer to your question. But this is not a "trivial exercise".
However, if you have wild sphere $S subset mathbbR^n$, then you get an obvious embedding of the suspension $Sigma S$ into $mathbbR^n+1$.
If $mathbbR^n backslash S$ is not uniformly locally 1-connected (1-ULC, see Rushing), then one can show that $mathbbR^n+1 backslash Sigma S$ also fails to be 1-ULC. Thus $Sigma S subset mathbbR^n+1$ is again a wild sphere. This applies to the horned sphere.
answered Sep 2 at 13:05
Paul Frost
4,838424
add a comment |Â
up vote
0
down vote
I recommend to have a look into
Rushing, T. Benny. Topological embeddings. Vol. 52. Academic Press, 1973.
Theorem 2.6.1 gives a positive answer to your question. But this is not a "trivial exercise".
However, if you have wild sphere $S subset mathbbR^n$, then you get an obvious embedding of the suspension $Sigma S$ into $mathbbR^n+1$.
If $mathbbR^n backslash S$ is not uniformly locally 1-connected (1-ULC, see Rushing), then one can show that $mathbbR^n+1 backslash Sigma S$ also fails to be 1-ULC. Thus $Sigma S subset mathbbR^n+1$ is again a wild sphere. This applies to the horned sphere.
answered Sep 2 at 13:05
Paul Frost
4,838424
add a comment |Â
up vote
0
down vote
up vote
0
down vote
I recommend to have a look into
Rushing, T. Benny. Topological embeddings. Vol. 52. Academic Press, 1973.
Theorem 2.6.1 gives a positive answer to your question. But this is not a "trivial exercise".
However, if you have wild sphere $S subset mathbbR^n$, then you get an obvious embedding of the suspension $Sigma S$ into $mathbbR^n+1$.
If $mathbbR^n backslash S$ is not uniformly locally 1-connected (1-ULC, see Rushing), then one can show that $mathbbR^n+1 backslash Sigma S$ also fails to be 1-ULC. Thus $Sigma S subset mathbbR^n+1$ is again a wild sphere. This applies to the horned sphere.
answered Sep 2 at 13:05
Paul Frost
4,838424
I recommend to have a look into
Rushing, T. Benny. Topological embeddings. Vol. 52. Academic Press, 1973.
Theorem 2.6.1 gives a positive answer to your question. But this is not a "trivial exercise".
However, if you have wild sphere $S subset mathbbR^n$, then you get an obvious embedding of the suspension $Sigma S$ into $mathbbR^n+1$.
If $mathbbR^n backslash S$ is not uniformly locally 1-connected (1-ULC, see Rushing), then one can show that $mathbbR^n+1 backslash Sigma S$ also fails to be 1-ULC. Thus $Sigma S subset mathbbR^n+1$ is again a wild sphere. This applies to the horned sphere.
answered Sep 2 at 13:05
Paul Frost
4,838424
answered Sep 2 at 13:05
Paul Frost
4,838424
answered Sep 2 at 13:05
Paul Frost
4,838424
answered Sep 2 at 13:05
Paul Frost
4,838424
4,838424
add a comment |Â
add a comment |Â
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I don't think anything concerning Alexander's horned sphere is trivial...
– Lord Shark the Unknown
Sep 2 at 9:39