Stiefel-Whitney class of $O(2)$ and $SO(2)$
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$O(2)$ is an extension of $mathbb{Z}_2$ by $SO(2)$,
$$1to SO(2) to O(2)to mathbb{Z}_2 to 1$$
Suppose the generator of $SO(2)$ is $j$, and the generator of $mathbb{Z}_2$ is $r$, then $O(2)$ is generated by the pair $(j, r)$ with $rj=j^{-1} r$ and $r^2=1$. Let $w_i(G)$ be the $i$-th stiefel whitney class of the $G$-bundle.
Then what is the relation between $w_2(O(2))$ and $w_2(SO(2))$? Are they the same?
If we instead of considering $O(2)$, but consider the group $Pin(2)^-$, which is generated by $(j, r)$ with $rj=j^{-1}r$, $r^2 j= j r^2$ and $r^4=1$,
then what is the relation between $w_2(Pin(2)^-)$ and $w_2(SO(2))$?
general-topology algebraic-topology differential-topology
edited Jan 5 at 22:07
wonderich
2,08631230
asked Nov 6 '18 at 20:41
user34104user34104
1478
|
show 1 more comment
$O(2)$ is an extension of $mathbb{Z}_2$ by $SO(2)$,
$$1to SO(2) to O(2)to mathbb{Z}_2 to 1$$
Suppose the generator of $SO(2)$ is $j$, and the generator of $mathbb{Z}_2$ is $r$, then $O(2)$ is generated by the pair $(j, r)$ with $rj=j^{-1} r$ and $r^2=1$. Let $w_i(G)$ be the $i$-th stiefel whitney class of the $G$-bundle.
Then what is the relation between $w_2(O(2))$ and $w_2(SO(2))$? Are they the same?
If we instead of considering $O(2)$, but consider the group $Pin(2)^-$, which is generated by $(j, r)$ with $rj=j^{-1}r$, $r^2 j= j r^2$ and $r^4=1$,
then what is the relation between $w_2(Pin(2)^-)$ and $w_2(SO(2))$?
general-topology algebraic-topology differential-topology
edited Jan 5 at 22:07
wonderich
2,08631230
asked Nov 6 '18 at 20:41
user34104user34104
1478
When you write "the $G$-bundle", what do you mean?
– Jason DeVito
Nov 6 '18 at 20:46
@JasonDeVito I mean principle bundle with structure group $G$, the usual definition in the standard construction in gauge theory.
– user34104
Nov 6 '18 at 20:53
I am not familiar with gauge theory. I've also never seen the notation $w_2(G)$ used for a principal $G$-bundle $Prightarrow B$, only $w_2(P)$. You also haven't mentioned how "the $SO(2)$-bundle" and "the $O(2)$-bundle" are related, so here is a guess: You have a principal $O(2)$ bundle $Qrightarrow B$ where $Q$ happens to have the form $Ptimes_{SO(2)} O(2)$ for some principal $SO(2)$-bundle $Prightarrow B$. And you're asking how $w_2(P),w_2(Q)in H^2(B;mathbb{Z}/2mathbb{Z})$ compare. Is this an accurate restatement of your first question?
– Jason DeVito
Nov 6 '18 at 21:12
@JasonDeVito Thanks for the clarification. Yes, this is what I mean.
– user34104
Nov 6 '18 at 21:15
I don't have time to write a full answer now (and I don't know the answer in the $Pin$ case), but for the case I just described above, we should have $w_2(P) = w_2(Q)$. Shortly, the classifying map $phi_P:Brightarrow BSO(2)$ for the $P$ bundle above should be the lift of the classifying map $phi_Q:Brightarrow BO(2)$. It is known (though don't know of a reference right off hand) that the map $H^ast(BO(n);mathbb{Z}/2mathbb{Z})rightarrow H^ast(BSO(n);mathbb{Z}/2mathbb{Z})$ maps $w_1 in H^1(BO(n);mathbb{Z}/2mathbb{Z})$ to $0$, but otherwise is the "identity map".
– Jason DeVito
Nov 6 '18 at 21:21
|
show 1 more comment
$O(2)$ is an extension of $mathbb{Z}_2$ by $SO(2)$,
$$1to SO(2) to O(2)to mathbb{Z}_2 to 1$$
Suppose the generator of $SO(2)$ is $j$, and the generator of $mathbb{Z}_2$ is $r$, then $O(2)$ is generated by the pair $(j, r)$ with $rj=j^{-1} r$ and $r^2=1$. Let $w_i(G)$ be the $i$-th stiefel whitney class of the $G$-bundle.
Then what is the relation between $w_2(O(2))$ and $w_2(SO(2))$? Are they the same?
If we instead of considering $O(2)$, but consider the group $Pin(2)^-$, which is generated by $(j, r)$ with $rj=j^{-1}r$, $r^2 j= j r^2$ and $r^4=1$,
then what is the relation between $w_2(Pin(2)^-)$ and $w_2(SO(2))$?
general-topology algebraic-topology differential-topology
edited Jan 5 at 22:07
wonderich
2,08631230
asked Nov 6 '18 at 20:41
user34104user34104
1478
$O(2)$ is an extension of $mathbb{Z}_2$ by $SO(2)$,
$$1to SO(2) to O(2)to mathbb{Z}_2 to 1$$
Suppose the generator of $SO(2)$ is $j$, and the generator of $mathbb{Z}_2$ is $r$, then $O(2)$ is generated by the pair $(j, r)$ with $rj=j^{-1} r$ and $r^2=1$. Let $w_i(G)$ be the $i$-th stiefel whitney class of the $G$-bundle.
Then what is the relation between $w_2(O(2))$ and $w_2(SO(2))$? Are they the same?
If we instead of considering $O(2)$, but consider the group $Pin(2)^-$, which is generated by $(j, r)$ with $rj=j^{-1}r$, $r^2 j= j r^2$ and $r^4=1$,
then what is the relation between $w_2(Pin(2)^-)$ and $w_2(SO(2))$?
general-topology algebraic-topology differential-topology
general-topology algebraic-topology differential-topology
edited Jan 5 at 22:07
wonderich
2,08631230
asked Nov 6 '18 at 20:41
user34104user34104
1478
edited Jan 5 at 22:07
wonderich
2,08631230
asked Nov 6 '18 at 20:41
user34104user34104
1478
edited Jan 5 at 22:07
wonderich
2,08631230
edited Jan 5 at 22:07
wonderich
2,08631230
edited Jan 5 at 22:07
wonderich
2,08631230
2,08631230
asked Nov 6 '18 at 20:41
user34104user34104
1478
asked Nov 6 '18 at 20:41
user34104user34104
1478
asked Nov 6 '18 at 20:41
user34104user34104
1478
1478
When you write "the $G$-bundle", what do you mean?
– Jason DeVito
Nov 6 '18 at 20:46
@JasonDeVito I mean principle bundle with structure group $G$, the usual definition in the standard construction in gauge theory.
– user34104
Nov 6 '18 at 20:53
I am not familiar with gauge theory. I've also never seen the notation $w_2(G)$ used for a principal $G$-bundle $Prightarrow B$, only $w_2(P)$. You also haven't mentioned how "the $SO(2)$-bundle" and "the $O(2)$-bundle" are related, so here is a guess: You have a principal $O(2)$ bundle $Qrightarrow B$ where $Q$ happens to have the form $Ptimes_{SO(2)} O(2)$ for some principal $SO(2)$-bundle $Prightarrow B$. And you're asking how $w_2(P),w_2(Q)in H^2(B;mathbb{Z}/2mathbb{Z})$ compare. Is this an accurate restatement of your first question?
– Jason DeVito
Nov 6 '18 at 21:12
@JasonDeVito Thanks for the clarification. Yes, this is what I mean.
– user34104
Nov 6 '18 at 21:15
I don't have time to write a full answer now (and I don't know the answer in the $Pin$ case), but for the case I just described above, we should have $w_2(P) = w_2(Q)$. Shortly, the classifying map $phi_P:Brightarrow BSO(2)$ for the $P$ bundle above should be the lift of the classifying map $phi_Q:Brightarrow BO(2)$. It is known (though don't know of a reference right off hand) that the map $H^ast(BO(n);mathbb{Z}/2mathbb{Z})rightarrow H^ast(BSO(n);mathbb{Z}/2mathbb{Z})$ maps $w_1 in H^1(BO(n);mathbb{Z}/2mathbb{Z})$ to $0$, but otherwise is the "identity map".
– Jason DeVito
Nov 6 '18 at 21:21
|
show 1 more comment
When you write "the $G$-bundle", what do you mean?
– Jason DeVito
Nov 6 '18 at 20:46
@JasonDeVito I mean principle bundle with structure group $G$, the usual definition in the standard construction in gauge theory.
– user34104
Nov 6 '18 at 20:53
I am not familiar with gauge theory. I've also never seen the notation $w_2(G)$ used for a principal $G$-bundle $Prightarrow B$, only $w_2(P)$. You also haven't mentioned how "the $SO(2)$-bundle" and "the $O(2)$-bundle" are related, so here is a guess: You have a principal $O(2)$ bundle $Qrightarrow B$ where $Q$ happens to have the form $Ptimes_{SO(2)} O(2)$ for some principal $SO(2)$-bundle $Prightarrow B$. And you're asking how $w_2(P),w_2(Q)in H^2(B;mathbb{Z}/2mathbb{Z})$ compare. Is this an accurate restatement of your first question?
– Jason DeVito
Nov 6 '18 at 21:12
@JasonDeVito Thanks for the clarification. Yes, this is what I mean.
– user34104
Nov 6 '18 at 21:15
I don't have time to write a full answer now (and I don't know the answer in the $Pin$ case), but for the case I just described above, we should have $w_2(P) = w_2(Q)$. Shortly, the classifying map $phi_P:Brightarrow BSO(2)$ for the $P$ bundle above should be the lift of the classifying map $phi_Q:Brightarrow BO(2)$. It is known (though don't know of a reference right off hand) that the map $H^ast(BO(n);mathbb{Z}/2mathbb{Z})rightarrow H^ast(BSO(n);mathbb{Z}/2mathbb{Z})$ maps $w_1 in H^1(BO(n);mathbb{Z}/2mathbb{Z})$ to $0$, but otherwise is the "identity map".
– Jason DeVito
Nov 6 '18 at 21:21
When you write "the $G$-bundle", what do you mean?
– Jason DeVito
Nov 6 '18 at 20:46
When you write "the $G$-bundle", what do you mean?
– Jason DeVito
Nov 6 '18 at 20:46
@JasonDeVito I mean principle bundle with structure group $G$, the usual definition in the standard construction in gauge theory.
– user34104
Nov 6 '18 at 20:53
@JasonDeVito I mean principle bundle with structure group $G$, the usual definition in the standard construction in gauge theory.
– user34104
Nov 6 '18 at 20:53
I am not familiar with gauge theory. I've also never seen the notation $w_2(G)$ used for a principal $G$-bundle $Prightarrow B$, only $w_2(P)$. You also haven't mentioned how "the $SO(2)$-bundle" and "the $O(2)$-bundle" are related, so here is a guess: You have a principal $O(2)$ bundle $Qrightarrow B$ where $Q$ happens to have the form $Ptimes_{SO(2)} O(2)$ for some principal $SO(2)$-bundle $Prightarrow B$. And you're asking how $w_2(P),w_2(Q)in H^2(B;mathbb{Z}/2mathbb{Z})$ compare. Is this an accurate restatement of your first question?
– Jason DeVito
Nov 6 '18 at 21:12
I am not familiar with gauge theory. I've also never seen the notation $w_2(G)$ used for a principal $G$-bundle $Prightarrow B$, only $w_2(P)$. You also haven't mentioned how "the $SO(2)$-bundle" and "the $O(2)$-bundle" are related, so here is a guess: You have a principal $O(2)$ bundle $Qrightarrow B$ where $Q$ happens to have the form $Ptimes_{SO(2)} O(2)$ for some principal $SO(2)$-bundle $Prightarrow B$. And you're asking how $w_2(P),w_2(Q)in H^2(B;mathbb{Z}/2mathbb{Z})$ compare. Is this an accurate restatement of your first question?
– Jason DeVito
Nov 6 '18 at 21:12
@JasonDeVito Thanks for the clarification. Yes, this is what I mean.
– user34104
Nov 6 '18 at 21:15
@JasonDeVito Thanks for the clarification. Yes, this is what I mean.
– user34104
Nov 6 '18 at 21:15
I don't have time to write a full answer now (and I don't know the answer in the $Pin$ case), but for the case I just described above, we should have $w_2(P) = w_2(Q)$. Shortly, the classifying map $phi_P:Brightarrow BSO(2)$ for the $P$ bundle above should be the lift of the classifying map $phi_Q:Brightarrow BO(2)$. It is known (though don't know of a reference right off hand) that the map $H^ast(BO(n);mathbb{Z}/2mathbb{Z})rightarrow H^ast(BSO(n);mathbb{Z}/2mathbb{Z})$ maps $w_1 in H^1(BO(n);mathbb{Z}/2mathbb{Z})$ to $0$, but otherwise is the "identity map".
– Jason DeVito
Nov 6 '18 at 21:21
I don't have time to write a full answer now (and I don't know the answer in the $Pin$ case), but for the case I just described above, we should have $w_2(P) = w_2(Q)$. Shortly, the classifying map $phi_P:Brightarrow BSO(2)$ for the $P$ bundle above should be the lift of the classifying map $phi_Q:Brightarrow BO(2)$. It is known (though don't know of a reference right off hand) that the map $H^ast(BO(n);mathbb{Z}/2mathbb{Z})rightarrow H^ast(BSO(n);mathbb{Z}/2mathbb{Z})$ maps $w_1 in H^1(BO(n);mathbb{Z}/2mathbb{Z})$ to $0$, but otherwise is the "identity map".
– Jason DeVito
Nov 6 '18 at 21:21
|
show 1 more comment
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When you write "the $G$-bundle", what do you mean?
– Jason DeVito
Nov 6 '18 at 20:46
@JasonDeVito I mean principle bundle with structure group $G$, the usual definition in the standard construction in gauge theory.
– user34104
Nov 6 '18 at 20:53
I am not familiar with gauge theory. I've also never seen the notation $w_2(G)$ used for a principal $G$-bundle $Prightarrow B$, only $w_2(P)$. You also haven't mentioned how "the $SO(2)$-bundle" and "the $O(2)$-bundle" are related, so here is a guess: You have a principal $O(2)$ bundle $Qrightarrow B$ where $Q$ happens to have the form $Ptimes_{SO(2)} O(2)$ for some principal $SO(2)$-bundle $Prightarrow B$. And you're asking how $w_2(P),w_2(Q)in H^2(B;mathbb{Z}/2mathbb{Z})$ compare. Is this an accurate restatement of your first question?
– Jason DeVito
Nov 6 '18 at 21:12
@JasonDeVito Thanks for the clarification. Yes, this is what I mean.
– user34104
Nov 6 '18 at 21:15
I don't have time to write a full answer now (and I don't know the answer in the $Pin$ case), but for the case I just described above, we should have $w_2(P) = w_2(Q)$. Shortly, the classifying map $phi_P:Brightarrow BSO(2)$ for the $P$ bundle above should be the lift of the classifying map $phi_Q:Brightarrow BO(2)$. It is known (though don't know of a reference right off hand) that the map $H^ast(BO(n);mathbb{Z}/2mathbb{Z})rightarrow H^ast(BSO(n);mathbb{Z}/2mathbb{Z})$ maps $w_1 in H^1(BO(n);mathbb{Z}/2mathbb{Z})$ to $0$, but otherwise is the "identity map".
– Jason DeVito
Nov 6 '18 at 21:21