“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

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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

share|cite|improve this question

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 | 

up vote
4
down vote

favorite

1

up vote
4
down vote

favorite

1

1

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

share|cite|improve this question

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

share|cite|improve this question

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.

share|cite|improve this question

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.

share|cite|improve this question

share|cite|improve this question

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.

add a comment | 

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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.

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answered 1 hour ago

Wlod AA

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1 Answer
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active

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1 Answer
1

active

oldest

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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.

share|cite|improve this answer

answered 1 hour ago

Wlod AA

1,236213

add a comment | 

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.

share|cite|improve this answer

answered 1 hour ago

Wlod AA

1,236213

add a comment | 

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.

share|cite|improve this answer

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.

share|cite|improve this answer

answered 1 hour ago

Wlod AA

1,236213

share|cite|improve this answer

share|cite|improve this answer

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 | 

add a comment | 

Douglas Sirk is a new contributor. Be nice, and check out our Code of Conduct.

 

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