What is the precise definition of a multi-connected manifold?
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I am looking for a term to describe manifolds that are connected but not simply connected. Multi-connected looks like a strong candidate. However, I can't seem to find a formal definition of the concept. What is the precise definition of a multi-connected manifold?
general-topology terminology
asked Dec 17 '18 at 0:01
AsdfAsdf
417213
$endgroup$
add a comment |
$begingroup$
I am looking for a term to describe manifolds that are connected but not simply connected. Multi-connected looks like a strong candidate. However, I can't seem to find a formal definition of the concept. What is the precise definition of a multi-connected manifold?
general-topology terminology
asked Dec 17 '18 at 0:01
AsdfAsdf
417213
$endgroup$
2
$begingroup$
You already gave it: a manifold that's connected but not simply connected. I think this is old language, though.
$endgroup$
– Qiaochu Yuan
Dec 17 '18 at 0:07
$begingroup$
How would you term it?
$endgroup$
– Asdf
Dec 17 '18 at 0:14
2
$begingroup$
Just "connected but not simply connected." It's not a concept I have much of a need for so I'm okay not having a term for it.
$endgroup$
– Qiaochu Yuan
Dec 17 '18 at 0:15
$begingroup$
Alright. I'll take multi-connected.
$endgroup$
– Asdf
Dec 17 '18 at 0:18
add a comment |
$begingroup$
I am looking for a term to describe manifolds that are connected but not simply connected. Multi-connected looks like a strong candidate. However, I can't seem to find a formal definition of the concept. What is the precise definition of a multi-connected manifold?
general-topology terminology
asked Dec 17 '18 at 0:01
AsdfAsdf
417213
$endgroup$
I am looking for a term to describe manifolds that are connected but not simply connected. Multi-connected looks like a strong candidate. However, I can't seem to find a formal definition of the concept. What is the precise definition of a multi-connected manifold?
general-topology terminology
general-topology terminology
asked Dec 17 '18 at 0:01
AsdfAsdf
417213
asked Dec 17 '18 at 0:01
AsdfAsdf
417213
asked Dec 17 '18 at 0:01
AsdfAsdf
417213
asked Dec 17 '18 at 0:01
AsdfAsdf
417213
asked Dec 17 '18 at 0:01
AsdfAsdf
417213
417213
2
$begingroup$
You already gave it: a manifold that's connected but not simply connected. I think this is old language, though.
$endgroup$
– Qiaochu Yuan
Dec 17 '18 at 0:07
$begingroup$
How would you term it?
$endgroup$
– Asdf
Dec 17 '18 at 0:14
2
$begingroup$
Just "connected but not simply connected." It's not a concept I have much of a need for so I'm okay not having a term for it.
$endgroup$
– Qiaochu Yuan
Dec 17 '18 at 0:15
$begingroup$
Alright. I'll take multi-connected.
$endgroup$
– Asdf
Dec 17 '18 at 0:18
add a comment |
2
$begingroup$
You already gave it: a manifold that's connected but not simply connected. I think this is old language, though.
$endgroup$
– Qiaochu Yuan
Dec 17 '18 at 0:07
$begingroup$
How would you term it?
$endgroup$
– Asdf
Dec 17 '18 at 0:14
2
$begingroup$
Just "connected but not simply connected." It's not a concept I have much of a need for so I'm okay not having a term for it.
$endgroup$
– Qiaochu Yuan
Dec 17 '18 at 0:15
$begingroup$
Alright. I'll take multi-connected.
$endgroup$
– Asdf
Dec 17 '18 at 0:18
2
2
$begingroup$
You already gave it: a manifold that's connected but not simply connected. I think this is old language, though.
$endgroup$
– Qiaochu Yuan
Dec 17 '18 at 0:07
$begingroup$
You already gave it: a manifold that's connected but not simply connected. I think this is old language, though.
$endgroup$
– Qiaochu Yuan
Dec 17 '18 at 0:07
$begingroup$
How would you term it?
$endgroup$
– Asdf
Dec 17 '18 at 0:14
$begingroup$
How would you term it?
$endgroup$
– Asdf
Dec 17 '18 at 0:14
2
2
$begingroup$
Just "connected but not simply connected." It's not a concept I have much of a need for so I'm okay not having a term for it.
$endgroup$
– Qiaochu Yuan
Dec 17 '18 at 0:15
$begingroup$
Just "connected but not simply connected." It's not a concept I have much of a need for so I'm okay not having a term for it.
$endgroup$
– Qiaochu Yuan
Dec 17 '18 at 0:15
$begingroup$
Alright. I'll take multi-connected.
$endgroup$
– Asdf
Dec 17 '18 at 0:18
$begingroup$
Alright. I'll take multi-connected.
$endgroup$
– Asdf
Dec 17 '18 at 0:18
add a comment |
2 Answers
2
active
oldest
votes
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I am not convinced that the expression "multi-connected" is a good choice for a connected but non-simply connected manifold.
The expression "doubly connected" is frequently used in complex analysis. A doubly connected region in the complex plane is a region bounded by two Jordan curves. See for example Conformal map of doubly connected domain into annulus..
A generalization is "multiply connected". See https://www.encyclopediaofmath.org/index.php/Multiply-connected_domain. Here are some references.
Walsh, J. L. "On the conformal mapping of multiply connected regions." Transactions of the American Mathematical Society 82.1 (1956): 128-146.
https://www.ams.org/journals/tran/1956-082-01/S0002-9947-1956-0080727-2/S0002-9947-1956-0080727-2.pdf
Walsh, J. L., and H. J. Landau. "On canonical conformal maps of multiply connected regions." Transactions of the American Mathematical Society 93.1 (1959): 81-96.
https://www.ams.org/journals/tran/1959-093-01/S0002-9947-1959-0160884-2/S0002-9947-1959-0160884-2.pdf
Landau, H. J. "On canonical conformal maps of multiply connected domains." Transactions of the American Mathematical Society 99.1 (1961): 1-20.
https://www.ams.org/journals/tran/1961-099-01/S0002-9947-1961-0121474-X/S0002-9947-1961-0121474-X.pdf
answered Dec 17 '18 at 14:29
Paul FrostPaul Frost
10.7k3933
$endgroup$
$begingroup$
Multiply connected looks like well-established terminology. It has a Mathworld page mathworld.wolfram.com/MultiplyConnected.html I found multi-connected in some physics literature, but it bothered me that it did not seem to turn up in the mathematics literature.
$endgroup$
– Asdf
Dec 18 '18 at 5:33
add a comment |
$begingroup$
A manifold that is connected (so path-connected too) but not simply connected, i.e. $pi_1(X,x_0)$ is not trivial (for some choice of base point, it matters not which, by path-connectedness).
answered Dec 17 '18 at 10:56
Henno BrandsmaHenno Brandsma
109k347115
$endgroup$
add a comment |
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2 Answers
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2 Answers
2
active
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$begingroup$
I am not convinced that the expression "multi-connected" is a good choice for a connected but non-simply connected manifold.
The expression "doubly connected" is frequently used in complex analysis. A doubly connected region in the complex plane is a region bounded by two Jordan curves. See for example Conformal map of doubly connected domain into annulus..
A generalization is "multiply connected". See https://www.encyclopediaofmath.org/index.php/Multiply-connected_domain. Here are some references.
Walsh, J. L. "On the conformal mapping of multiply connected regions." Transactions of the American Mathematical Society 82.1 (1956): 128-146.
https://www.ams.org/journals/tran/1956-082-01/S0002-9947-1956-0080727-2/S0002-9947-1956-0080727-2.pdf
Walsh, J. L., and H. J. Landau. "On canonical conformal maps of multiply connected regions." Transactions of the American Mathematical Society 93.1 (1959): 81-96.
https://www.ams.org/journals/tran/1959-093-01/S0002-9947-1959-0160884-2/S0002-9947-1959-0160884-2.pdf
Landau, H. J. "On canonical conformal maps of multiply connected domains." Transactions of the American Mathematical Society 99.1 (1961): 1-20.
https://www.ams.org/journals/tran/1961-099-01/S0002-9947-1961-0121474-X/S0002-9947-1961-0121474-X.pdf
answered Dec 17 '18 at 14:29
Paul FrostPaul Frost
10.7k3933
$endgroup$
$begingroup$
Multiply connected looks like well-established terminology. It has a Mathworld page mathworld.wolfram.com/MultiplyConnected.html I found multi-connected in some physics literature, but it bothered me that it did not seem to turn up in the mathematics literature.
$endgroup$
– Asdf
Dec 18 '18 at 5:33
add a comment |
$begingroup$
I am not convinced that the expression "multi-connected" is a good choice for a connected but non-simply connected manifold.
The expression "doubly connected" is frequently used in complex analysis. A doubly connected region in the complex plane is a region bounded by two Jordan curves. See for example Conformal map of doubly connected domain into annulus..
A generalization is "multiply connected". See https://www.encyclopediaofmath.org/index.php/Multiply-connected_domain. Here are some references.
Walsh, J. L. "On the conformal mapping of multiply connected regions." Transactions of the American Mathematical Society 82.1 (1956): 128-146.
https://www.ams.org/journals/tran/1956-082-01/S0002-9947-1956-0080727-2/S0002-9947-1956-0080727-2.pdf
Walsh, J. L., and H. J. Landau. "On canonical conformal maps of multiply connected regions." Transactions of the American Mathematical Society 93.1 (1959): 81-96.
https://www.ams.org/journals/tran/1959-093-01/S0002-9947-1959-0160884-2/S0002-9947-1959-0160884-2.pdf
Landau, H. J. "On canonical conformal maps of multiply connected domains." Transactions of the American Mathematical Society 99.1 (1961): 1-20.
https://www.ams.org/journals/tran/1961-099-01/S0002-9947-1961-0121474-X/S0002-9947-1961-0121474-X.pdf
answered Dec 17 '18 at 14:29
Paul FrostPaul Frost
10.7k3933
$endgroup$
$begingroup$
Multiply connected looks like well-established terminology. It has a Mathworld page mathworld.wolfram.com/MultiplyConnected.html I found multi-connected in some physics literature, but it bothered me that it did not seem to turn up in the mathematics literature.
$endgroup$
– Asdf
Dec 18 '18 at 5:33
add a comment |
$begingroup$
I am not convinced that the expression "multi-connected" is a good choice for a connected but non-simply connected manifold.
The expression "doubly connected" is frequently used in complex analysis. A doubly connected region in the complex plane is a region bounded by two Jordan curves. See for example Conformal map of doubly connected domain into annulus..
A generalization is "multiply connected". See https://www.encyclopediaofmath.org/index.php/Multiply-connected_domain. Here are some references.
Walsh, J. L. "On the conformal mapping of multiply connected regions." Transactions of the American Mathematical Society 82.1 (1956): 128-146.
https://www.ams.org/journals/tran/1956-082-01/S0002-9947-1956-0080727-2/S0002-9947-1956-0080727-2.pdf
Walsh, J. L., and H. J. Landau. "On canonical conformal maps of multiply connected regions." Transactions of the American Mathematical Society 93.1 (1959): 81-96.
https://www.ams.org/journals/tran/1959-093-01/S0002-9947-1959-0160884-2/S0002-9947-1959-0160884-2.pdf
Landau, H. J. "On canonical conformal maps of multiply connected domains." Transactions of the American Mathematical Society 99.1 (1961): 1-20.
https://www.ams.org/journals/tran/1961-099-01/S0002-9947-1961-0121474-X/S0002-9947-1961-0121474-X.pdf
answered Dec 17 '18 at 14:29
Paul FrostPaul Frost
10.7k3933
$endgroup$
I am not convinced that the expression "multi-connected" is a good choice for a connected but non-simply connected manifold.
The expression "doubly connected" is frequently used in complex analysis. A doubly connected region in the complex plane is a region bounded by two Jordan curves. See for example Conformal map of doubly connected domain into annulus..
A generalization is "multiply connected". See https://www.encyclopediaofmath.org/index.php/Multiply-connected_domain. Here are some references.
Walsh, J. L. "On the conformal mapping of multiply connected regions." Transactions of the American Mathematical Society 82.1 (1956): 128-146.
https://www.ams.org/journals/tran/1956-082-01/S0002-9947-1956-0080727-2/S0002-9947-1956-0080727-2.pdf
Walsh, J. L., and H. J. Landau. "On canonical conformal maps of multiply connected regions." Transactions of the American Mathematical Society 93.1 (1959): 81-96.
https://www.ams.org/journals/tran/1959-093-01/S0002-9947-1959-0160884-2/S0002-9947-1959-0160884-2.pdf
Landau, H. J. "On canonical conformal maps of multiply connected domains." Transactions of the American Mathematical Society 99.1 (1961): 1-20.
https://www.ams.org/journals/tran/1961-099-01/S0002-9947-1961-0121474-X/S0002-9947-1961-0121474-X.pdf
answered Dec 17 '18 at 14:29
Paul FrostPaul Frost
10.7k3933
answered Dec 17 '18 at 14:29
Paul FrostPaul Frost
10.7k3933
answered Dec 17 '18 at 14:29
Paul FrostPaul Frost
10.7k3933
answered Dec 17 '18 at 14:29
Paul FrostPaul Frost
10.7k3933
10.7k3933
$begingroup$
Multiply connected looks like well-established terminology. It has a Mathworld page mathworld.wolfram.com/MultiplyConnected.html I found multi-connected in some physics literature, but it bothered me that it did not seem to turn up in the mathematics literature.
$endgroup$
– Asdf
Dec 18 '18 at 5:33
add a comment |
$begingroup$
Multiply connected looks like well-established terminology. It has a Mathworld page mathworld.wolfram.com/MultiplyConnected.html I found multi-connected in some physics literature, but it bothered me that it did not seem to turn up in the mathematics literature.
$endgroup$
– Asdf
Dec 18 '18 at 5:33
$begingroup$
Multiply connected looks like well-established terminology. It has a Mathworld page mathworld.wolfram.com/MultiplyConnected.html I found multi-connected in some physics literature, but it bothered me that it did not seem to turn up in the mathematics literature.
$endgroup$
– Asdf
Dec 18 '18 at 5:33
$begingroup$
Multiply connected looks like well-established terminology. It has a Mathworld page mathworld.wolfram.com/MultiplyConnected.html I found multi-connected in some physics literature, but it bothered me that it did not seem to turn up in the mathematics literature.
$endgroup$
– Asdf
Dec 18 '18 at 5:33
add a comment |
$begingroup$
A manifold that is connected (so path-connected too) but not simply connected, i.e. $pi_1(X,x_0)$ is not trivial (for some choice of base point, it matters not which, by path-connectedness).
answered Dec 17 '18 at 10:56
Henno BrandsmaHenno Brandsma
109k347115
$endgroup$
add a comment |
$begingroup$
A manifold that is connected (so path-connected too) but not simply connected, i.e. $pi_1(X,x_0)$ is not trivial (for some choice of base point, it matters not which, by path-connectedness).
answered Dec 17 '18 at 10:56
Henno BrandsmaHenno Brandsma
109k347115
$endgroup$
add a comment |
$begingroup$
A manifold that is connected (so path-connected too) but not simply connected, i.e. $pi_1(X,x_0)$ is not trivial (for some choice of base point, it matters not which, by path-connectedness).
answered Dec 17 '18 at 10:56
Henno BrandsmaHenno Brandsma
109k347115
$endgroup$
A manifold that is connected (so path-connected too) but not simply connected, i.e. $pi_1(X,x_0)$ is not trivial (for some choice of base point, it matters not which, by path-connectedness).
answered Dec 17 '18 at 10:56
Henno BrandsmaHenno Brandsma
109k347115
answered Dec 17 '18 at 10:56
Henno BrandsmaHenno Brandsma
109k347115
answered Dec 17 '18 at 10:56
Henno BrandsmaHenno Brandsma
109k347115
answered Dec 17 '18 at 10:56
Henno BrandsmaHenno Brandsma
109k347115
109k347115
add a comment |
add a comment |
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2
$begingroup$
You already gave it: a manifold that's connected but not simply connected. I think this is old language, though.
$endgroup$
– Qiaochu Yuan
Dec 17 '18 at 0:07
$begingroup$
How would you term it?
$endgroup$
– Asdf
Dec 17 '18 at 0:14
2
$begingroup$
Just "connected but not simply connected." It's not a concept I have much of a need for so I'm okay not having a term for it.
$endgroup$
– Qiaochu Yuan
Dec 17 '18 at 0:15
$begingroup$
Alright. I'll take multi-connected.
$endgroup$
– Asdf
Dec 17 '18 at 0:18