“Cyclic†continuum
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On p. 221 of http://topology.auburn.edu/tp/reprints/v08/tp08113.pdf, I found the following definition:
"A curve is said to be cyclic if its first Čech cohomology group with integer coefficients does not vanish".
Here, a curve means a homogeneous metric continuum of dimension 1.
Can someone explain this definition in different, more elementary, terms, and give some examples to illustrate the meaning?
gn.general-topology cohomology definitions
edited Sep 30 at 22:27
Amir Sagiv
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up vote
6
down vote
favorite
On p. 221 of http://topology.auburn.edu/tp/reprints/v08/tp08113.pdf, I found the following definition:
"A curve is said to be cyclic if its first Čech cohomology group with integer coefficients does not vanish".
Here, a curve means a homogeneous metric continuum of dimension 1.
Can someone explain this definition in different, more elementary, terms, and give some examples to illustrate the meaning?
gn.general-topology cohomology definitions
edited Sep 30 at 22:27
Amir Sagiv
1,7151127
add a comment |Â
up vote
6
down vote
favorite
up vote
6
down vote
favorite
On p. 221 of http://topology.auburn.edu/tp/reprints/v08/tp08113.pdf, I found the following definition:
"A curve is said to be cyclic if its first Čech cohomology group with integer coefficients does not vanish".
Here, a curve means a homogeneous metric continuum of dimension 1.
Can someone explain this definition in different, more elementary, terms, and give some examples to illustrate the meaning?
gn.general-topology cohomology definitions
edited Sep 30 at 22:27
Amir Sagiv
1,7151127
On p. 221 of http://topology.auburn.edu/tp/reprints/v08/tp08113.pdf, I found the following definition:
"A curve is said to be cyclic if its first Čech cohomology group with integer coefficients does not vanish".
Here, a curve means a homogeneous metric continuum of dimension 1.
Can someone explain this definition in different, more elementary, terms, and give some examples to illustrate the meaning?
gn.general-topology cohomology definitions
gn.general-topology cohomology definitions
edited Sep 30 at 22:27
Amir Sagiv
1,7151127
edited Sep 30 at 22:27
Amir Sagiv
1,7151127
edited Sep 30 at 22:27
Amir Sagiv
1,7151127
edited Sep 30 at 22:27
Amir Sagiv
1,7151127
edited Sep 30 at 22:27
Amir Sagiv
1,7151127
1,7151127
asked Sep 30 at 21:09
user129590
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1 Answer
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Perhaps the first Čech cohomology group with integer coefficients may seem not elementary. Then use the characterization of it as the first Borsuk's cohomotopy group (or it appears as the Brushlinsky's group in the well-known text on Homotopy Theory by Hu). This group's elements are simply the homotopy classes of mappings into $ mathbb S^1.$ Space $ S^1 $ is a topological group, it induces the group structure on the homotopy classes.
answered Sep 30 at 21:22
Wlod AA
1,256214
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
Perhaps the first Čech cohomology group with integer coefficients may seem not elementary. Then use the characterization of it as the first Borsuk's cohomotopy group (or it appears as the Brushlinsky's group in the well-known text on Homotopy Theory by Hu). This group's elements are simply the homotopy classes of mappings into $ mathbb S^1.$ Space $ S^1 $ is a topological group, it induces the group structure on the homotopy classes.
answered Sep 30 at 21:22
Wlod AA
1,256214
add a comment |Â
up vote
5
down vote
Perhaps the first Čech cohomology group with integer coefficients may seem not elementary. Then use the characterization of it as the first Borsuk's cohomotopy group (or it appears as the Brushlinsky's group in the well-known text on Homotopy Theory by Hu). This group's elements are simply the homotopy classes of mappings into $ mathbb S^1.$ Space $ S^1 $ is a topological group, it induces the group structure on the homotopy classes.
answered Sep 30 at 21:22
Wlod AA
1,256214
add a comment |Â
up vote
5
down vote
up vote
5
down vote
Perhaps the first Čech cohomology group with integer coefficients may seem not elementary. Then use the characterization of it as the first Borsuk's cohomotopy group (or it appears as the Brushlinsky's group in the well-known text on Homotopy Theory by Hu). This group's elements are simply the homotopy classes of mappings into $ mathbb S^1.$ Space $ S^1 $ is a topological group, it induces the group structure on the homotopy classes.
answered Sep 30 at 21:22
Wlod AA
1,256214
Perhaps the first Čech cohomology group with integer coefficients may seem not elementary. Then use the characterization of it as the first Borsuk's cohomotopy group (or it appears as the Brushlinsky's group in the well-known text on Homotopy Theory by Hu). This group's elements are simply the homotopy classes of mappings into $ mathbb S^1.$ Space $ S^1 $ is a topological group, it induces the group structure on the homotopy classes.
answered Sep 30 at 21:22
Wlod AA
1,256214
answered Sep 30 at 21:22
Wlod AA
1,256214
answered Sep 30 at 21:22
Wlod AA
1,256214
answered Sep 30 at 21:22
Wlod AA
1,256214
1,256214
add a comment |Â
add a comment |Â
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