Mixing
“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 4 mins ago
Amir Sagiv
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Douglas Sirk
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up vote
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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 4 mins ago
Amir Sagiv
1,7171126
asked 1 hour ago
Douglas Sirk
211
New contributor
Douglas Sirk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
4
down vote
favorite
up vote
4
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 4 mins ago
Amir Sagiv
1,7171126
asked 1 hour ago
Douglas Sirk
211
New contributor
Douglas Sirk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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 4 mins ago
Amir Sagiv
1,7171126
asked 1 hour ago
Douglas Sirk
211
New contributor
Douglas Sirk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 4 mins ago
Amir Sagiv
1,7171126
asked 1 hour ago
Douglas Sirk
211
New contributor
Douglas Sirk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 4 mins ago
Amir Sagiv
1,7171126
edited 4 mins ago
Amir Sagiv
1,7171126
edited 4 mins ago
Amir Sagiv
1,7171126
1,7171126
asked 1 hour ago
Douglas Sirk
211
New contributor
Douglas Sirk 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
Douglas Sirk
211
asked 1 hour ago
Douglas Sirk
211
211
New contributor
Douglas Sirk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Douglas Sirk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Douglas Sirk 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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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 1 hour ago
Wlod AA
1,236213
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
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 1 hour ago
Wlod AA
1,236213
add a comment |Â
up vote
3
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 1 hour ago
Wlod AA
1,236213
add a comment |Â
up vote
3
down vote
up vote
3
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 1 hour ago
Wlod AA
1,236213
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 1 hour ago
Wlod AA
1,236213
answered 1 hour ago
Wlod AA
1,236213
answered 1 hour ago
Wlod AA
1,236213
answered 1 hour ago
Wlod AA
1,236213
1,236213
add a comment |Â
add a comment |Â
Douglas Sirk is a new contributor. Be nice, and check out our Code of Conduct.
Douglas Sirk is a new contributor. Be nice, and check out our Code of Conduct.
Douglas Sirk is a new contributor. Be nice, and check out our Code of Conduct.
Douglas Sirk is a new contributor. Be nice, and check out our Code of Conduct.
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