Compute $pi_2(mathbb{S}^2,X)$ where $X$ is the figure 8
$begingroup$
We have the following short exact sequence from the long exact sequence for a pair
$$0topi_2(mathbb{S^2})=mathbb{Z}topi_2(mathbb{S}^2,X)topi_1(X)=F_2to0.$$
I wanted to construct a section (I guess there is one), so $pi_2(mathbb{S}^2)=mathbb{Z}rtimes F_2$. What I did is that I tried to use the homotopy extension property for $(mathbb{D}^2,mathbb{S}^1)$ to say that I can map a homotopy class in $pi_1(X)$ to maps that are still homotopic in the set of maps $mathbb{D}^2tomathbb{S}^2$ and whose restrictions to $mathbb{S}^1$ are also homotopic, so I get a section.
But say, the section $s:pi_1(X)topi_2(mathbb{S}^2,X)$ maps a class $[f]inpi_1(X)$ to $[F]inpi_2(mathbb{S}^2,X)$ where $F$ is an extension of $f$. Later I think not every map in $[F]$ restricted to $mathbb{S}^1$ is homotopic to $f$... And if the homotopy extension property really works, the boundary map would always have a section, which seems absurd, though I am totally new to the relative homotopy groups... Even if the group is really a semidirect product, I have no idea how to figure out the action of $F_2$ on $mathbb{Z}$. My guess is something like $varphiintext{Aut}(mathbb{Z})$, $varphi(*^{d_1}*^{d_2}cdots*^{d_k})=sum d_i$, where $*^{d_1}*^{d_2}cdots*^{d_k}$ denotes a word in $F_2$.
Hope to get some help... Thanks!
algebraic-topology
asked Dec 7 '18 at 2:23
chikurinchikurin
899
$endgroup$
add a comment |
$begingroup$
We have the following short exact sequence from the long exact sequence for a pair
$$0topi_2(mathbb{S^2})=mathbb{Z}topi_2(mathbb{S}^2,X)topi_1(X)=F_2to0.$$
I wanted to construct a section (I guess there is one), so $pi_2(mathbb{S}^2)=mathbb{Z}rtimes F_2$. What I did is that I tried to use the homotopy extension property for $(mathbb{D}^2,mathbb{S}^1)$ to say that I can map a homotopy class in $pi_1(X)$ to maps that are still homotopic in the set of maps $mathbb{D}^2tomathbb{S}^2$ and whose restrictions to $mathbb{S}^1$ are also homotopic, so I get a section.
But say, the section $s:pi_1(X)topi_2(mathbb{S}^2,X)$ maps a class $[f]inpi_1(X)$ to $[F]inpi_2(mathbb{S}^2,X)$ where $F$ is an extension of $f$. Later I think not every map in $[F]$ restricted to $mathbb{S}^1$ is homotopic to $f$... And if the homotopy extension property really works, the boundary map would always have a section, which seems absurd, though I am totally new to the relative homotopy groups... Even if the group is really a semidirect product, I have no idea how to figure out the action of $F_2$ on $mathbb{Z}$. My guess is something like $varphiintext{Aut}(mathbb{Z})$, $varphi(*^{d_1}*^{d_2}cdots*^{d_k})=sum d_i$, where $*^{d_1}*^{d_2}cdots*^{d_k}$ denotes a word in $F_2$.
Hope to get some help... Thanks!
algebraic-topology
asked Dec 7 '18 at 2:23
chikurinchikurin
899
$endgroup$
add a comment |
$begingroup$
We have the following short exact sequence from the long exact sequence for a pair
$$0topi_2(mathbb{S^2})=mathbb{Z}topi_2(mathbb{S}^2,X)topi_1(X)=F_2to0.$$
I wanted to construct a section (I guess there is one), so $pi_2(mathbb{S}^2)=mathbb{Z}rtimes F_2$. What I did is that I tried to use the homotopy extension property for $(mathbb{D}^2,mathbb{S}^1)$ to say that I can map a homotopy class in $pi_1(X)$ to maps that are still homotopic in the set of maps $mathbb{D}^2tomathbb{S}^2$ and whose restrictions to $mathbb{S}^1$ are also homotopic, so I get a section.
But say, the section $s:pi_1(X)topi_2(mathbb{S}^2,X)$ maps a class $[f]inpi_1(X)$ to $[F]inpi_2(mathbb{S}^2,X)$ where $F$ is an extension of $f$. Later I think not every map in $[F]$ restricted to $mathbb{S}^1$ is homotopic to $f$... And if the homotopy extension property really works, the boundary map would always have a section, which seems absurd, though I am totally new to the relative homotopy groups... Even if the group is really a semidirect product, I have no idea how to figure out the action of $F_2$ on $mathbb{Z}$. My guess is something like $varphiintext{Aut}(mathbb{Z})$, $varphi(*^{d_1}*^{d_2}cdots*^{d_k})=sum d_i$, where $*^{d_1}*^{d_2}cdots*^{d_k}$ denotes a word in $F_2$.
Hope to get some help... Thanks!
algebraic-topology
asked Dec 7 '18 at 2:23
chikurinchikurin
899
$endgroup$
We have the following short exact sequence from the long exact sequence for a pair
$$0topi_2(mathbb{S^2})=mathbb{Z}topi_2(mathbb{S}^2,X)topi_1(X)=F_2to0.$$
I wanted to construct a section (I guess there is one), so $pi_2(mathbb{S}^2)=mathbb{Z}rtimes F_2$. What I did is that I tried to use the homotopy extension property for $(mathbb{D}^2,mathbb{S}^1)$ to say that I can map a homotopy class in $pi_1(X)$ to maps that are still homotopic in the set of maps $mathbb{D}^2tomathbb{S}^2$ and whose restrictions to $mathbb{S}^1$ are also homotopic, so I get a section.
But say, the section $s:pi_1(X)topi_2(mathbb{S}^2,X)$ maps a class $[f]inpi_1(X)$ to $[F]inpi_2(mathbb{S}^2,X)$ where $F$ is an extension of $f$. Later I think not every map in $[F]$ restricted to $mathbb{S}^1$ is homotopic to $f$... And if the homotopy extension property really works, the boundary map would always have a section, which seems absurd, though I am totally new to the relative homotopy groups... Even if the group is really a semidirect product, I have no idea how to figure out the action of $F_2$ on $mathbb{Z}$. My guess is something like $varphiintext{Aut}(mathbb{Z})$, $varphi(*^{d_1}*^{d_2}cdots*^{d_k})=sum d_i$, where $*^{d_1}*^{d_2}cdots*^{d_k}$ denotes a word in $F_2$.
Hope to get some help... Thanks!
algebraic-topology
algebraic-topology
asked Dec 7 '18 at 2:23
chikurinchikurin
899
asked Dec 7 '18 at 2:23
chikurinchikurin
899
edited Dec 7 '18 at 2:29
chikurin
asked Dec 7 '18 at 2:23
chikurinchikurin
899
asked Dec 7 '18 at 2:23
chikurinchikurin
899
asked Dec 7 '18 at 2:23
chikurinchikurin
899
899
add a comment |
add a comment |
1 Answer
1
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oldest
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$begingroup$
Since $pi_2(A)$ is central in $pi_2(B,A)$, it is never the nontrivial semidirect product. [See Spanier, Algebraic Topology, Chapter 7, Section 3, Corollary 13, page 386]
The fact that your sequence splits follows formally because $F_2$ is a free group.
Since you mentioned the more general question of the sequence splitting, if we have an exact sequence
$$0 to pi_2(A) to pi_2(A,X) to pi_1(X) to 0$$
I think we can say the sequence is split if $X to A$ is nulhomotopic ("$X$ is nulhomotopic in $A$"). This way you can choose a nulhomotopy and use it to make a filling-in of each path, and also a homotopy of their homotopies.
I think this agrees with Tyler Lawson's comment here https://mathoverflow.net/questions/294006/conditions-for-the-second-homotopy-group-to-be-abelian
answered Dec 7 '18 at 8:28
BenBen
3,118616
$endgroup$
1
$begingroup$
@chikurin No problem :) your last question is a little out of my depth in topology but I'll suggest a line of thought. There is a functor from pairs of pointed spaces to crossed modules (maybe its the same as the one taking the central extension above to the corresponding crossed module) - cf. answers posted on MSE by Ronnie Brown. So you can first find a central extension crossed module that is not split, and try to represent it topologically.
$endgroup$
– Ben
Dec 9 '18 at 11:41
1
$begingroup$
Also, just to check - the question your asking is: does there exist a pair of spaces $X to S$ (with $X$ a subspace) such that $pi_1(X) to pi_1(S)$ is the zero map and $pi_2(S,X) to pi_1(X)$ is split? In particular you don't insist $S = S^2$ I think.
$endgroup$
– Ben
Dec 9 '18 at 11:47
1
$begingroup$
@chikurin One problem in the above line is that you will have to find out the corresponding condition to $pi_1(X to S^2) = 0$ for crossed modules, or modify your problem to ask when $pi_2(S,X) to text{im}(delta)$ is split.
$endgroup$
– Ben
Dec 9 '18 at 11:49
1
$begingroup$
@chikurin Finally, it's worth mentioning that there is a correspondence between 2-types and crossed modules (and strict 2-groupoids and weak 2-groupoids). This might be relevant, but I don't know how compatible it is with the relation to pairs of spaces from above.
$endgroup$
– Ben
Dec 9 '18 at 11:51
1
$begingroup$
@chikurin I haven't studied crossed modules in any depth, so if you are stuck, you also might consider asking this as a separate question as it will be more likely to get attention from the good topologists on here.
$endgroup$
– Ben
Dec 9 '18 at 11:54
|
show 1 more comment
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$begingroup$
Since $pi_2(A)$ is central in $pi_2(B,A)$, it is never the nontrivial semidirect product. [See Spanier, Algebraic Topology, Chapter 7, Section 3, Corollary 13, page 386]
The fact that your sequence splits follows formally because $F_2$ is a free group.
Since you mentioned the more general question of the sequence splitting, if we have an exact sequence
$$0 to pi_2(A) to pi_2(A,X) to pi_1(X) to 0$$
I think we can say the sequence is split if $X to A$ is nulhomotopic ("$X$ is nulhomotopic in $A$"). This way you can choose a nulhomotopy and use it to make a filling-in of each path, and also a homotopy of their homotopies.
I think this agrees with Tyler Lawson's comment here https://mathoverflow.net/questions/294006/conditions-for-the-second-homotopy-group-to-be-abelian
answered Dec 7 '18 at 8:28
BenBen
3,118616
$endgroup$
1
$begingroup$
@chikurin No problem :) your last question is a little out of my depth in topology but I'll suggest a line of thought. There is a functor from pairs of pointed spaces to crossed modules (maybe its the same as the one taking the central extension above to the corresponding crossed module) - cf. answers posted on MSE by Ronnie Brown. So you can first find a central extension crossed module that is not split, and try to represent it topologically.
$endgroup$
– Ben
Dec 9 '18 at 11:41
1
$begingroup$
Also, just to check - the question your asking is: does there exist a pair of spaces $X to S$ (with $X$ a subspace) such that $pi_1(X) to pi_1(S)$ is the zero map and $pi_2(S,X) to pi_1(X)$ is split? In particular you don't insist $S = S^2$ I think.
$endgroup$
– Ben
Dec 9 '18 at 11:47
1
$begingroup$
@chikurin One problem in the above line is that you will have to find out the corresponding condition to $pi_1(X to S^2) = 0$ for crossed modules, or modify your problem to ask when $pi_2(S,X) to text{im}(delta)$ is split.
$endgroup$
– Ben
Dec 9 '18 at 11:49
1
$begingroup$
@chikurin Finally, it's worth mentioning that there is a correspondence between 2-types and crossed modules (and strict 2-groupoids and weak 2-groupoids). This might be relevant, but I don't know how compatible it is with the relation to pairs of spaces from above.
$endgroup$
– Ben
Dec 9 '18 at 11:51
1
$begingroup$
@chikurin I haven't studied crossed modules in any depth, so if you are stuck, you also might consider asking this as a separate question as it will be more likely to get attention from the good topologists on here.
$endgroup$
– Ben
Dec 9 '18 at 11:54
|
show 1 more comment
$begingroup$
Since $pi_2(A)$ is central in $pi_2(B,A)$, it is never the nontrivial semidirect product. [See Spanier, Algebraic Topology, Chapter 7, Section 3, Corollary 13, page 386]
The fact that your sequence splits follows formally because $F_2$ is a free group.
Since you mentioned the more general question of the sequence splitting, if we have an exact sequence
$$0 to pi_2(A) to pi_2(A,X) to pi_1(X) to 0$$
I think we can say the sequence is split if $X to A$ is nulhomotopic ("$X$ is nulhomotopic in $A$"). This way you can choose a nulhomotopy and use it to make a filling-in of each path, and also a homotopy of their homotopies.
I think this agrees with Tyler Lawson's comment here https://mathoverflow.net/questions/294006/conditions-for-the-second-homotopy-group-to-be-abelian
answered Dec 7 '18 at 8:28
BenBen
3,118616
$endgroup$
1
$begingroup$
@chikurin No problem :) your last question is a little out of my depth in topology but I'll suggest a line of thought. There is a functor from pairs of pointed spaces to crossed modules (maybe its the same as the one taking the central extension above to the corresponding crossed module) - cf. answers posted on MSE by Ronnie Brown. So you can first find a central extension crossed module that is not split, and try to represent it topologically.
$endgroup$
– Ben
Dec 9 '18 at 11:41
1
$begingroup$
Also, just to check - the question your asking is: does there exist a pair of spaces $X to S$ (with $X$ a subspace) such that $pi_1(X) to pi_1(S)$ is the zero map and $pi_2(S,X) to pi_1(X)$ is split? In particular you don't insist $S = S^2$ I think.
$endgroup$
– Ben
Dec 9 '18 at 11:47
1
$begingroup$
@chikurin One problem in the above line is that you will have to find out the corresponding condition to $pi_1(X to S^2) = 0$ for crossed modules, or modify your problem to ask when $pi_2(S,X) to text{im}(delta)$ is split.
$endgroup$
– Ben
Dec 9 '18 at 11:49
1
$begingroup$
@chikurin Finally, it's worth mentioning that there is a correspondence between 2-types and crossed modules (and strict 2-groupoids and weak 2-groupoids). This might be relevant, but I don't know how compatible it is with the relation to pairs of spaces from above.
$endgroup$
– Ben
Dec 9 '18 at 11:51
1
$begingroup$
@chikurin I haven't studied crossed modules in any depth, so if you are stuck, you also might consider asking this as a separate question as it will be more likely to get attention from the good topologists on here.
$endgroup$
– Ben
Dec 9 '18 at 11:54
|
show 1 more comment
$begingroup$
Since $pi_2(A)$ is central in $pi_2(B,A)$, it is never the nontrivial semidirect product. [See Spanier, Algebraic Topology, Chapter 7, Section 3, Corollary 13, page 386]
The fact that your sequence splits follows formally because $F_2$ is a free group.
Since you mentioned the more general question of the sequence splitting, if we have an exact sequence
$$0 to pi_2(A) to pi_2(A,X) to pi_1(X) to 0$$
I think we can say the sequence is split if $X to A$ is nulhomotopic ("$X$ is nulhomotopic in $A$"). This way you can choose a nulhomotopy and use it to make a filling-in of each path, and also a homotopy of their homotopies.
I think this agrees with Tyler Lawson's comment here https://mathoverflow.net/questions/294006/conditions-for-the-second-homotopy-group-to-be-abelian
answered Dec 7 '18 at 8:28
BenBen
3,118616
$endgroup$
Since $pi_2(A)$ is central in $pi_2(B,A)$, it is never the nontrivial semidirect product. [See Spanier, Algebraic Topology, Chapter 7, Section 3, Corollary 13, page 386]
The fact that your sequence splits follows formally because $F_2$ is a free group.
Since you mentioned the more general question of the sequence splitting, if we have an exact sequence
$$0 to pi_2(A) to pi_2(A,X) to pi_1(X) to 0$$
I think we can say the sequence is split if $X to A$ is nulhomotopic ("$X$ is nulhomotopic in $A$"). This way you can choose a nulhomotopy and use it to make a filling-in of each path, and also a homotopy of their homotopies.
I think this agrees with Tyler Lawson's comment here https://mathoverflow.net/questions/294006/conditions-for-the-second-homotopy-group-to-be-abelian
answered Dec 7 '18 at 8:28
BenBen
3,118616
answered Dec 7 '18 at 8:28
BenBen
3,118616
answered Dec 7 '18 at 8:28
BenBen
3,118616
answered Dec 7 '18 at 8:28
BenBen
3,118616
3,118616
1
$begingroup$
@chikurin No problem :) your last question is a little out of my depth in topology but I'll suggest a line of thought. There is a functor from pairs of pointed spaces to crossed modules (maybe its the same as the one taking the central extension above to the corresponding crossed module) - cf. answers posted on MSE by Ronnie Brown. So you can first find a central extension crossed module that is not split, and try to represent it topologically.
$endgroup$
– Ben
Dec 9 '18 at 11:41
1
$begingroup$
Also, just to check - the question your asking is: does there exist a pair of spaces $X to S$ (with $X$ a subspace) such that $pi_1(X) to pi_1(S)$ is the zero map and $pi_2(S,X) to pi_1(X)$ is split? In particular you don't insist $S = S^2$ I think.
$endgroup$
– Ben
Dec 9 '18 at 11:47
1
$begingroup$
@chikurin One problem in the above line is that you will have to find out the corresponding condition to $pi_1(X to S^2) = 0$ for crossed modules, or modify your problem to ask when $pi_2(S,X) to text{im}(delta)$ is split.
$endgroup$
– Ben
Dec 9 '18 at 11:49
1
$begingroup$
@chikurin Finally, it's worth mentioning that there is a correspondence between 2-types and crossed modules (and strict 2-groupoids and weak 2-groupoids). This might be relevant, but I don't know how compatible it is with the relation to pairs of spaces from above.
$endgroup$
– Ben
Dec 9 '18 at 11:51
1
$begingroup$
@chikurin I haven't studied crossed modules in any depth, so if you are stuck, you also might consider asking this as a separate question as it will be more likely to get attention from the good topologists on here.
$endgroup$
– Ben
Dec 9 '18 at 11:54
|
show 1 more comment
1
$begingroup$
@chikurin No problem :) your last question is a little out of my depth in topology but I'll suggest a line of thought. There is a functor from pairs of pointed spaces to crossed modules (maybe its the same as the one taking the central extension above to the corresponding crossed module) - cf. answers posted on MSE by Ronnie Brown. So you can first find a central extension crossed module that is not split, and try to represent it topologically.
$endgroup$
– Ben
Dec 9 '18 at 11:41
1
$begingroup$
Also, just to check - the question your asking is: does there exist a pair of spaces $X to S$ (with $X$ a subspace) such that $pi_1(X) to pi_1(S)$ is the zero map and $pi_2(S,X) to pi_1(X)$ is split? In particular you don't insist $S = S^2$ I think.
$endgroup$
– Ben
Dec 9 '18 at 11:47
1
$begingroup$
@chikurin One problem in the above line is that you will have to find out the corresponding condition to $pi_1(X to S^2) = 0$ for crossed modules, or modify your problem to ask when $pi_2(S,X) to text{im}(delta)$ is split.
$endgroup$
– Ben
Dec 9 '18 at 11:49
1
$begingroup$
@chikurin Finally, it's worth mentioning that there is a correspondence between 2-types and crossed modules (and strict 2-groupoids and weak 2-groupoids). This might be relevant, but I don't know how compatible it is with the relation to pairs of spaces from above.
$endgroup$
– Ben
Dec 9 '18 at 11:51
1
$begingroup$
@chikurin I haven't studied crossed modules in any depth, so if you are stuck, you also might consider asking this as a separate question as it will be more likely to get attention from the good topologists on here.
$endgroup$
– Ben
Dec 9 '18 at 11:54
1
1
$begingroup$
@chikurin No problem :) your last question is a little out of my depth in topology but I'll suggest a line of thought. There is a functor from pairs of pointed spaces to crossed modules (maybe its the same as the one taking the central extension above to the corresponding crossed module) - cf. answers posted on MSE by Ronnie Brown. So you can first find a central extension crossed module that is not split, and try to represent it topologically.
$endgroup$
– Ben
Dec 9 '18 at 11:41
$begingroup$
@chikurin No problem :) your last question is a little out of my depth in topology but I'll suggest a line of thought. There is a functor from pairs of pointed spaces to crossed modules (maybe its the same as the one taking the central extension above to the corresponding crossed module) - cf. answers posted on MSE by Ronnie Brown. So you can first find a central extension crossed module that is not split, and try to represent it topologically.
$endgroup$
– Ben
Dec 9 '18 at 11:41
1
1
$begingroup$
Also, just to check - the question your asking is: does there exist a pair of spaces $X to S$ (with $X$ a subspace) such that $pi_1(X) to pi_1(S)$ is the zero map and $pi_2(S,X) to pi_1(X)$ is split? In particular you don't insist $S = S^2$ I think.
$endgroup$
– Ben
Dec 9 '18 at 11:47
$begingroup$
Also, just to check - the question your asking is: does there exist a pair of spaces $X to S$ (with $X$ a subspace) such that $pi_1(X) to pi_1(S)$ is the zero map and $pi_2(S,X) to pi_1(X)$ is split? In particular you don't insist $S = S^2$ I think.
$endgroup$
– Ben
Dec 9 '18 at 11:47
1
1
$begingroup$
@chikurin One problem in the above line is that you will have to find out the corresponding condition to $pi_1(X to S^2) = 0$ for crossed modules, or modify your problem to ask when $pi_2(S,X) to text{im}(delta)$ is split.
$endgroup$
– Ben
Dec 9 '18 at 11:49
$begingroup$
@chikurin One problem in the above line is that you will have to find out the corresponding condition to $pi_1(X to S^2) = 0$ for crossed modules, or modify your problem to ask when $pi_2(S,X) to text{im}(delta)$ is split.
$endgroup$
– Ben
Dec 9 '18 at 11:49
1
1
$begingroup$
@chikurin Finally, it's worth mentioning that there is a correspondence between 2-types and crossed modules (and strict 2-groupoids and weak 2-groupoids). This might be relevant, but I don't know how compatible it is with the relation to pairs of spaces from above.
$endgroup$
– Ben
Dec 9 '18 at 11:51
$begingroup$
@chikurin Finally, it's worth mentioning that there is a correspondence between 2-types and crossed modules (and strict 2-groupoids and weak 2-groupoids). This might be relevant, but I don't know how compatible it is with the relation to pairs of spaces from above.
$endgroup$
– Ben
Dec 9 '18 at 11:51
1
1
$begingroup$
@chikurin I haven't studied crossed modules in any depth, so if you are stuck, you also might consider asking this as a separate question as it will be more likely to get attention from the good topologists on here.
$endgroup$
– Ben
Dec 9 '18 at 11:54
$begingroup$
@chikurin I haven't studied crossed modules in any depth, so if you are stuck, you also might consider asking this as a separate question as it will be more likely to get attention from the good topologists on here.
$endgroup$
– Ben
Dec 9 '18 at 11:54
|
show 1 more comment
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