Attaching $2$-dimensional cell to $D^2$ gives the space $S^2/(xsim -x)$
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I am studying Algebraic Topology, and right now I am going through cell-attachment, which I have a pretty hard time to grasp. An "example" they give in the book is:
Example:
Define $X$ to be the space obtained from $S^2$ by identifying antipodal points on the equator, then it is easy to see that $X$ can be obtained by attaching a $2$-dimensional cell to $D^2$.
I suppose this should be easy, but it isn't for me, so I would be really happy if someone could help me through this example and how to define the map. Since I don't even know where (and how) to start.
general-topology algebraic-topology cw-complexes
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add a comment |
$begingroup$
I am studying Algebraic Topology, and right now I am going through cell-attachment, which I have a pretty hard time to grasp. An "example" they give in the book is:
Example:
Define $X$ to be the space obtained from $S^2$ by identifying antipodal points on the equator, then it is easy to see that $X$ can be obtained by attaching a $2$-dimensional cell to $D^2$.
I suppose this should be easy, but it isn't for me, so I would be really happy if someone could help me through this example and how to define the map. Since I don't even know where (and how) to start.
general-topology algebraic-topology cw-complexes
$endgroup$
add a comment |
$begingroup$
I am studying Algebraic Topology, and right now I am going through cell-attachment, which I have a pretty hard time to grasp. An "example" they give in the book is:
Example:
Define $X$ to be the space obtained from $S^2$ by identifying antipodal points on the equator, then it is easy to see that $X$ can be obtained by attaching a $2$-dimensional cell to $D^2$.
I suppose this should be easy, but it isn't for me, so I would be really happy if someone could help me through this example and how to define the map. Since I don't even know where (and how) to start.
general-topology algebraic-topology cw-complexes
$endgroup$
I am studying Algebraic Topology, and right now I am going through cell-attachment, which I have a pretty hard time to grasp. An "example" they give in the book is:
Example:
Define $X$ to be the space obtained from $S^2$ by identifying antipodal points on the equator, then it is easy to see that $X$ can be obtained by attaching a $2$-dimensional cell to $D^2$.
I suppose this should be easy, but it isn't for me, so I would be really happy if someone could help me through this example and how to define the map. Since I don't even know where (and how) to start.
general-topology algebraic-topology cw-complexes
general-topology algebraic-topology cw-complexes
asked Dec 10 '17 at 0:01
user511893
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1 Answer
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Take $D^2$ and let the attaching map be $e:partial D^2 to partial D^2$ be the quotient map, as in $z mapsto z^2 $ as complex numbers.
Also note that if the attaching map were identity, we would just recover $S^2$.
answered Dec 10 '17 at 0:21
Andres MejiaAndres Mejia
16.2k21548
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when you attach a cell, you attach its boundary.
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– Tsemo Aristide
Dec 10 '17 at 0:28
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@TsemoAristide my apologies, i both misread the question and said something silly.
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– Andres Mejia
Dec 10 '17 at 0:37
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Thank you for your answer Andres. This looks like an answer I am happy with. :) Anyhow, I will have to take a closer look at this tomorrow (and maybe also have some questions by then) since it's in the middle of the night here and I am starting to become... tired. ;)
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– user511893
Dec 10 '17 at 0:45
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@AndresMejia Probably a silly question, but why do you choose to map $-x$ to $x$ and not $x$ to $-x$? :)
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– user511893
Dec 10 '17 at 10:13
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Dear Andres, yes that's quite okay with me and I have deleted my now irrelevant previous comment.
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– Georges Elencwajg
Dec 31 '18 at 18:50
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Your Answer
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Take $D^2$ and let the attaching map be $e:partial D^2 to partial D^2$ be the quotient map, as in $z mapsto z^2 $ as complex numbers.
Also note that if the attaching map were identity, we would just recover $S^2$.
answered Dec 10 '17 at 0:21
Andres MejiaAndres Mejia
16.2k21548
$endgroup$
$begingroup$
when you attach a cell, you attach its boundary.
$endgroup$
– Tsemo Aristide
Dec 10 '17 at 0:28
$begingroup$
@TsemoAristide my apologies, i both misread the question and said something silly.
$endgroup$
– Andres Mejia
Dec 10 '17 at 0:37
$begingroup$
Thank you for your answer Andres. This looks like an answer I am happy with. :) Anyhow, I will have to take a closer look at this tomorrow (and maybe also have some questions by then) since it's in the middle of the night here and I am starting to become... tired. ;)
$endgroup$
– user511893
Dec 10 '17 at 0:45
$begingroup$
@AndresMejia Probably a silly question, but why do you choose to map $-x$ to $x$ and not $x$ to $-x$? :)
$endgroup$
– user511893
Dec 10 '17 at 10:13
$begingroup$
Dear Andres, yes that's quite okay with me and I have deleted my now irrelevant previous comment.
$endgroup$
– Georges Elencwajg
Dec 31 '18 at 18:50
add a comment |
$begingroup$
Take $D^2$ and let the attaching map be $e:partial D^2 to partial D^2$ be the quotient map, as in $z mapsto z^2 $ as complex numbers.
Also note that if the attaching map were identity, we would just recover $S^2$.
answered Dec 10 '17 at 0:21
Andres MejiaAndres Mejia
16.2k21548
$endgroup$
$begingroup$
when you attach a cell, you attach its boundary.
$endgroup$
– Tsemo Aristide
Dec 10 '17 at 0:28
$begingroup$
@TsemoAristide my apologies, i both misread the question and said something silly.
$endgroup$
– Andres Mejia
Dec 10 '17 at 0:37
$begingroup$
Thank you for your answer Andres. This looks like an answer I am happy with. :) Anyhow, I will have to take a closer look at this tomorrow (and maybe also have some questions by then) since it's in the middle of the night here and I am starting to become... tired. ;)
$endgroup$
– user511893
Dec 10 '17 at 0:45
$begingroup$
@AndresMejia Probably a silly question, but why do you choose to map $-x$ to $x$ and not $x$ to $-x$? :)
$endgroup$
– user511893
Dec 10 '17 at 10:13
$begingroup$
Dear Andres, yes that's quite okay with me and I have deleted my now irrelevant previous comment.
$endgroup$
– Georges Elencwajg
Dec 31 '18 at 18:50
add a comment |
$begingroup$
Take $D^2$ and let the attaching map be $e:partial D^2 to partial D^2$ be the quotient map, as in $z mapsto z^2 $ as complex numbers.
Also note that if the attaching map were identity, we would just recover $S^2$.
answered Dec 10 '17 at 0:21
Andres MejiaAndres Mejia
16.2k21548
$endgroup$
Take $D^2$ and let the attaching map be $e:partial D^2 to partial D^2$ be the quotient map, as in $z mapsto z^2 $ as complex numbers.
Also note that if the attaching map were identity, we would just recover $S^2$.
answered Dec 10 '17 at 0:21
Andres MejiaAndres Mejia
16.2k21548
edited Dec 31 '18 at 18:20
answered Dec 10 '17 at 0:21
Andres MejiaAndres Mejia
16.2k21548
answered Dec 10 '17 at 0:21
Andres MejiaAndres Mejia
16.2k21548
answered Dec 10 '17 at 0:21
Andres MejiaAndres Mejia
16.2k21548
16.2k21548
$begingroup$
when you attach a cell, you attach its boundary.
$endgroup$
– Tsemo Aristide
Dec 10 '17 at 0:28
$begingroup$
@TsemoAristide my apologies, i both misread the question and said something silly.
$endgroup$
– Andres Mejia
Dec 10 '17 at 0:37
$begingroup$
Thank you for your answer Andres. This looks like an answer I am happy with. :) Anyhow, I will have to take a closer look at this tomorrow (and maybe also have some questions by then) since it's in the middle of the night here and I am starting to become... tired. ;)
$endgroup$
– user511893
Dec 10 '17 at 0:45
$begingroup$
@AndresMejia Probably a silly question, but why do you choose to map $-x$ to $x$ and not $x$ to $-x$? :)
$endgroup$
– user511893
Dec 10 '17 at 10:13
$begingroup$
Dear Andres, yes that's quite okay with me and I have deleted my now irrelevant previous comment.
$endgroup$
– Georges Elencwajg
Dec 31 '18 at 18:50
add a comment |
$begingroup$
when you attach a cell, you attach its boundary.
$endgroup$
– Tsemo Aristide
Dec 10 '17 at 0:28
$begingroup$
@TsemoAristide my apologies, i both misread the question and said something silly.
$endgroup$
– Andres Mejia
Dec 10 '17 at 0:37
$begingroup$
Thank you for your answer Andres. This looks like an answer I am happy with. :) Anyhow, I will have to take a closer look at this tomorrow (and maybe also have some questions by then) since it's in the middle of the night here and I am starting to become... tired. ;)
$endgroup$
– user511893
Dec 10 '17 at 0:45
$begingroup$
@AndresMejia Probably a silly question, but why do you choose to map $-x$ to $x$ and not $x$ to $-x$? :)
$endgroup$
– user511893
Dec 10 '17 at 10:13
$begingroup$
Dear Andres, yes that's quite okay with me and I have deleted my now irrelevant previous comment.
$endgroup$
– Georges Elencwajg
Dec 31 '18 at 18:50
$begingroup$
when you attach a cell, you attach its boundary.
$endgroup$
– Tsemo Aristide
Dec 10 '17 at 0:28
$begingroup$
when you attach a cell, you attach its boundary.
$endgroup$
– Tsemo Aristide
Dec 10 '17 at 0:28
$begingroup$
@TsemoAristide my apologies, i both misread the question and said something silly.
$endgroup$
– Andres Mejia
Dec 10 '17 at 0:37
$begingroup$
@TsemoAristide my apologies, i both misread the question and said something silly.
$endgroup$
– Andres Mejia
Dec 10 '17 at 0:37
$begingroup$
Thank you for your answer Andres. This looks like an answer I am happy with. :) Anyhow, I will have to take a closer look at this tomorrow (and maybe also have some questions by then) since it's in the middle of the night here and I am starting to become... tired. ;)
$endgroup$
– user511893
Dec 10 '17 at 0:45
$begingroup$
Thank you for your answer Andres. This looks like an answer I am happy with. :) Anyhow, I will have to take a closer look at this tomorrow (and maybe also have some questions by then) since it's in the middle of the night here and I am starting to become... tired. ;)
$endgroup$
– user511893
Dec 10 '17 at 0:45
$begingroup$
@AndresMejia Probably a silly question, but why do you choose to map $-x$ to $x$ and not $x$ to $-x$? :)
$endgroup$
– user511893
Dec 10 '17 at 10:13
$begingroup$
@AndresMejia Probably a silly question, but why do you choose to map $-x$ to $x$ and not $x$ to $-x$? :)
$endgroup$
– user511893
Dec 10 '17 at 10:13
$begingroup$
Dear Andres, yes that's quite okay with me and I have deleted my now irrelevant previous comment.
$endgroup$
– Georges Elencwajg
Dec 31 '18 at 18:50
$begingroup$
Dear Andres, yes that's quite okay with me and I have deleted my now irrelevant previous comment.
$endgroup$
– Georges Elencwajg
Dec 31 '18 at 18:50
add a comment |
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