Example of a CW complex that is not a $Delta$-complex?
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Hatcher notes in chapter 2.1 (in the paragraph just preceding the section on simplicial homology (page 104 in my edition)), that all $Delta$-complexes can be realized as CW complexes with some added restrictions on the characteristic maps. Is there a simple example of a CW complex that is not a $Delta$-complex?
general-topology algebraic-topology cw-complexes
asked Aug 11 at 21:14
Arbutus
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Hatcher notes in chapter 2.1 (in the paragraph just preceding the section on simplicial homology (page 104 in my edition)), that all $Delta$-complexes can be realized as CW complexes with some added restrictions on the characteristic maps. Is there a simple example of a CW complex that is not a $Delta$-complex?
general-topology algebraic-topology cw-complexes
asked Aug 11 at 21:14
Arbutus
538515
1
Take the unique cell complex which has one $0$-cell and one $2$-cell.
– Cheerful Parsnip
Aug 11 at 21:18
1
I guess that makes sense. Can you explain a bit more? It seems like you've just described $S^2$. Is the idea basically that the restriction to each of its boundaries of the characteristic map of the 2-cell can't be a characteristic map for a 1-cell because there are no 1-cells? So would any space that admits a CW complex structure with an $n$-cell but no $(n-1)$-cell be an example?
– Arbutus
Aug 11 at 21:35
2
Yes, this is a cell decomposition of $S^2$, and yes any such similar example would work.
– Cheerful Parsnip
Aug 11 at 21:53
Great, thanks so much! If you want to write exactly your first comment as a solution I will accept it.
– Arbutus
Aug 11 at 22:31
I'm pretty sure every $Delta$ complex is at least homotopy equivalent to a CW complex
– leibnewtz
Aug 11 at 22:45
 |Â
show 1 more comment
up vote
1
down vote
favorite
1
up vote
1
down vote
favorite
1
1
Hatcher notes in chapter 2.1 (in the paragraph just preceding the section on simplicial homology (page 104 in my edition)), that all $Delta$-complexes can be realized as CW complexes with some added restrictions on the characteristic maps. Is there a simple example of a CW complex that is not a $Delta$-complex?
general-topology algebraic-topology cw-complexes
asked Aug 11 at 21:14
Arbutus
538515
Hatcher notes in chapter 2.1 (in the paragraph just preceding the section on simplicial homology (page 104 in my edition)), that all $Delta$-complexes can be realized as CW complexes with some added restrictions on the characteristic maps. Is there a simple example of a CW complex that is not a $Delta$-complex?
general-topology algebraic-topology cw-complexes
asked Aug 11 at 21:14
Arbutus
538515
edited Aug 11 at 22:44
asked Aug 11 at 21:14
Arbutus
538515
asked Aug 11 at 21:14
Arbutus
538515
asked Aug 11 at 21:14
Arbutus
538515
538515
1
Take the unique cell complex which has one $0$-cell and one $2$-cell.
– Cheerful Parsnip
Aug 11 at 21:18
1
I guess that makes sense. Can you explain a bit more? It seems like you've just described $S^2$. Is the idea basically that the restriction to each of its boundaries of the characteristic map of the 2-cell can't be a characteristic map for a 1-cell because there are no 1-cells? So would any space that admits a CW complex structure with an $n$-cell but no $(n-1)$-cell be an example?
– Arbutus
Aug 11 at 21:35
2
Yes, this is a cell decomposition of $S^2$, and yes any such similar example would work.
– Cheerful Parsnip
Aug 11 at 21:53
Great, thanks so much! If you want to write exactly your first comment as a solution I will accept it.
– Arbutus
Aug 11 at 22:31
I'm pretty sure every $Delta$ complex is at least homotopy equivalent to a CW complex
– leibnewtz
Aug 11 at 22:45
 |Â
show 1 more comment
1
Take the unique cell complex which has one $0$-cell and one $2$-cell.
– Cheerful Parsnip
Aug 11 at 21:18
1
I guess that makes sense. Can you explain a bit more? It seems like you've just described $S^2$. Is the idea basically that the restriction to each of its boundaries of the characteristic map of the 2-cell can't be a characteristic map for a 1-cell because there are no 1-cells? So would any space that admits a CW complex structure with an $n$-cell but no $(n-1)$-cell be an example?
– Arbutus
Aug 11 at 21:35
2
Yes, this is a cell decomposition of $S^2$, and yes any such similar example would work.
– Cheerful Parsnip
Aug 11 at 21:53
Great, thanks so much! If you want to write exactly your first comment as a solution I will accept it.
– Arbutus
Aug 11 at 22:31
I'm pretty sure every $Delta$ complex is at least homotopy equivalent to a CW complex
– leibnewtz
Aug 11 at 22:45
1
1
Take the unique cell complex which has one $0$-cell and one $2$-cell.
– Cheerful Parsnip
Aug 11 at 21:18
Take the unique cell complex which has one $0$-cell and one $2$-cell.
– Cheerful Parsnip
Aug 11 at 21:18
1
1
I guess that makes sense. Can you explain a bit more? It seems like you've just described $S^2$. Is the idea basically that the restriction to each of its boundaries of the characteristic map of the 2-cell can't be a characteristic map for a 1-cell because there are no 1-cells? So would any space that admits a CW complex structure with an $n$-cell but no $(n-1)$-cell be an example?
– Arbutus
Aug 11 at 21:35
I guess that makes sense. Can you explain a bit more? It seems like you've just described $S^2$. Is the idea basically that the restriction to each of its boundaries of the characteristic map of the 2-cell can't be a characteristic map for a 1-cell because there are no 1-cells? So would any space that admits a CW complex structure with an $n$-cell but no $(n-1)$-cell be an example?
– Arbutus
Aug 11 at 21:35
2
2
Yes, this is a cell decomposition of $S^2$, and yes any such similar example would work.
– Cheerful Parsnip
Aug 11 at 21:53
Yes, this is a cell decomposition of $S^2$, and yes any such similar example would work.
– Cheerful Parsnip
Aug 11 at 21:53
Great, thanks so much! If you want to write exactly your first comment as a solution I will accept it.
– Arbutus
Aug 11 at 22:31
Great, thanks so much! If you want to write exactly your first comment as a solution I will accept it.
– Arbutus
Aug 11 at 22:31
I'm pretty sure every $Delta$ complex is at least homotopy equivalent to a CW complex
– leibnewtz
Aug 11 at 22:45
I'm pretty sure every $Delta$ complex is at least homotopy equivalent to a CW complex
– leibnewtz
Aug 11 at 22:45
 |Â
show 1 more comment
1 Answer
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Take the unique cell complex which has one $0$-cell and one $2$-cell.
answered Aug 11 at 23:48
Cheerful Parsnip
20.4k23194
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Take the unique cell complex which has one $0$-cell and one $2$-cell.
answered Aug 11 at 23:48
Cheerful Parsnip
20.4k23194
add a comment |Â
up vote
2
down vote
accepted
Take the unique cell complex which has one $0$-cell and one $2$-cell.
answered Aug 11 at 23:48
Cheerful Parsnip
20.4k23194
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Take the unique cell complex which has one $0$-cell and one $2$-cell.
answered Aug 11 at 23:48
Cheerful Parsnip
20.4k23194
Take the unique cell complex which has one $0$-cell and one $2$-cell.
answered Aug 11 at 23:48
Cheerful Parsnip
20.4k23194
answered Aug 11 at 23:48
Cheerful Parsnip
20.4k23194
answered Aug 11 at 23:48
Cheerful Parsnip
20.4k23194
answered Aug 11 at 23:48
Cheerful Parsnip
20.4k23194
20.4k23194
add a comment |Â
add a comment |Â
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1
Take the unique cell complex which has one $0$-cell and one $2$-cell.
– Cheerful Parsnip
Aug 11 at 21:18
1
I guess that makes sense. Can you explain a bit more? It seems like you've just described $S^2$. Is the idea basically that the restriction to each of its boundaries of the characteristic map of the 2-cell can't be a characteristic map for a 1-cell because there are no 1-cells? So would any space that admits a CW complex structure with an $n$-cell but no $(n-1)$-cell be an example?
– Arbutus
Aug 11 at 21:35
2
Yes, this is a cell decomposition of $S^2$, and yes any such similar example would work.
– Cheerful Parsnip
Aug 11 at 21:53
Great, thanks so much! If you want to write exactly your first comment as a solution I will accept it.
– Arbutus
Aug 11 at 22:31
I'm pretty sure every $Delta$ complex is at least homotopy equivalent to a CW complex
– leibnewtz
Aug 11 at 22:45