Mixing
$H$ is admissible, then $Prightarrow P/H$ is a principle $H$ -bundle.
$begingroup$
Proposition 3.5, page 5: Suppose $P rightarrow B$ is a principal $G$-bundle, and let $H$ be an admissible subgroup of $G$. Then the quotient map $P rightarrow P/H$ is a principal $H$-bundle.
Definition: Call a subgroup $H$ of $G$ admissible if the quotient map $G rightarrow G/H$ is a principal $H$-bundle.
The proof given goes as follows: we can identify the map with the map,
$$P times_G G rightarrow P times_G (G/H)$$
and we utilize the fact that $H$ of $G$ is admissible.
Now I could not really spell out the maps of local trivilization, we want open set of $P times_G (G/H)$ such that its preimage, in $P times_G G$ has a local trivilization.
Justifying this seems like a mess. I would like to see some neat approach.
My attempt goes as follows.
Since the quotient map is an open map, take a local trivilization, open set $U$ of $G/H$, so $P times U$ has open image in $P times_G(G/H)$, where $pi:G rightarrow G/H$, under quotient $t:P times G/H rightarrow P times_G (G/H)$
Consider its preimage, which I believe should be the image of $P times pi^{-1}(U)$ under the quotient. $s:P times G rightarrow P times_G G$
So we show that we have a $G$-homeomorphism,
$$ s(P times pi^{-1}(U)) rightarrow t(P times U) times (P times_G G/H)$$
Everything here seems messy. But is this right?
algebraic-topology fiber-bundles principal-bundles
asked Jan 20 at 15:34
CL.CL.
2,2712925
$endgroup$
add a comment |
$begingroup$
Proposition 3.5, page 5: Suppose $P rightarrow B$ is a principal $G$-bundle, and let $H$ be an admissible subgroup of $G$. Then the quotient map $P rightarrow P/H$ is a principal $H$-bundle.
Definition: Call a subgroup $H$ of $G$ admissible if the quotient map $G rightarrow G/H$ is a principal $H$-bundle.
The proof given goes as follows: we can identify the map with the map,
$$P times_G G rightarrow P times_G (G/H)$$
and we utilize the fact that $H$ of $G$ is admissible.
Now I could not really spell out the maps of local trivilization, we want open set of $P times_G (G/H)$ such that its preimage, in $P times_G G$ has a local trivilization.
Justifying this seems like a mess. I would like to see some neat approach.
My attempt goes as follows.
Since the quotient map is an open map, take a local trivilization, open set $U$ of $G/H$, so $P times U$ has open image in $P times_G(G/H)$, where $pi:G rightarrow G/H$, under quotient $t:P times G/H rightarrow P times_G (G/H)$
Consider its preimage, which I believe should be the image of $P times pi^{-1}(U)$ under the quotient. $s:P times G rightarrow P times_G G$
So we show that we have a $G$-homeomorphism,
$$ s(P times pi^{-1}(U)) rightarrow t(P times U) times (P times_G G/H)$$
Everything here seems messy. But is this right?
algebraic-topology fiber-bundles principal-bundles
asked Jan 20 at 15:34
CL.CL.
2,2712925
$endgroup$
$begingroup$
what is your $Ptimes_GG$ and $Ptimes_{G/H}G/H$??
$endgroup$
– Praphulla Koushik
Jan 20 at 16:44
$begingroup$
A principal bundle is trivial if and only if it has a section. Take a trivialisation of $p:Prightarrow B$ over an open subset $Vsubseteq B$ and consider the image of $p^{-1}(V)times U$ in $Ptimes_GG/H$, where $U$ here is the same as you consider above.
$endgroup$
– Tyrone
Jan 21 at 9:55
$begingroup$
Thanks Tyrone, I will try spell this out, @Koushik, those are defined as in the notes, we quotient out by the diagonal action when $P times G$ is regarded as a $G$ space, i.e. $(pg,h) sim (p,gh)$. for $p in P, g,h in G$.
$endgroup$
– CL.
Jan 22 at 8:52
add a comment |
$begingroup$
Proposition 3.5, page 5: Suppose $P rightarrow B$ is a principal $G$-bundle, and let $H$ be an admissible subgroup of $G$. Then the quotient map $P rightarrow P/H$ is a principal $H$-bundle.
Definition: Call a subgroup $H$ of $G$ admissible if the quotient map $G rightarrow G/H$ is a principal $H$-bundle.
The proof given goes as follows: we can identify the map with the map,
$$P times_G G rightarrow P times_G (G/H)$$
and we utilize the fact that $H$ of $G$ is admissible.
Now I could not really spell out the maps of local trivilization, we want open set of $P times_G (G/H)$ such that its preimage, in $P times_G G$ has a local trivilization.
Justifying this seems like a mess. I would like to see some neat approach.
My attempt goes as follows.
Since the quotient map is an open map, take a local trivilization, open set $U$ of $G/H$, so $P times U$ has open image in $P times_G(G/H)$, where $pi:G rightarrow G/H$, under quotient $t:P times G/H rightarrow P times_G (G/H)$
Consider its preimage, which I believe should be the image of $P times pi^{-1}(U)$ under the quotient. $s:P times G rightarrow P times_G G$
So we show that we have a $G$-homeomorphism,
$$ s(P times pi^{-1}(U)) rightarrow t(P times U) times (P times_G G/H)$$
Everything here seems messy. But is this right?
algebraic-topology fiber-bundles principal-bundles
asked Jan 20 at 15:34
CL.CL.
2,2712925
$endgroup$
Proposition 3.5, page 5: Suppose $P rightarrow B$ is a principal $G$-bundle, and let $H$ be an admissible subgroup of $G$. Then the quotient map $P rightarrow P/H$ is a principal $H$-bundle.
Definition: Call a subgroup $H$ of $G$ admissible if the quotient map $G rightarrow G/H$ is a principal $H$-bundle.
The proof given goes as follows: we can identify the map with the map,
$$P times_G G rightarrow P times_G (G/H)$$
and we utilize the fact that $H$ of $G$ is admissible.
Now I could not really spell out the maps of local trivilization, we want open set of $P times_G (G/H)$ such that its preimage, in $P times_G G$ has a local trivilization.
Justifying this seems like a mess. I would like to see some neat approach.
My attempt goes as follows.
Since the quotient map is an open map, take a local trivilization, open set $U$ of $G/H$, so $P times U$ has open image in $P times_G(G/H)$, where $pi:G rightarrow G/H$, under quotient $t:P times G/H rightarrow P times_G (G/H)$
Consider its preimage, which I believe should be the image of $P times pi^{-1}(U)$ under the quotient. $s:P times G rightarrow P times_G G$
So we show that we have a $G$-homeomorphism,
$$ s(P times pi^{-1}(U)) rightarrow t(P times U) times (P times_G G/H)$$
Everything here seems messy. But is this right?
algebraic-topology fiber-bundles principal-bundles
algebraic-topology fiber-bundles principal-bundles
asked Jan 20 at 15:34
CL.CL.
2,2712925
asked Jan 20 at 15:34
CL.CL.
2,2712925
asked Jan 20 at 15:34
CL.CL.
2,2712925
asked Jan 20 at 15:34
CL.CL.
2,2712925
asked Jan 20 at 15:34
CL.CL.
2,2712925
2,2712925
$begingroup$
what is your $Ptimes_GG$ and $Ptimes_{G/H}G/H$??
$endgroup$
– Praphulla Koushik
Jan 20 at 16:44
$begingroup$
A principal bundle is trivial if and only if it has a section. Take a trivialisation of $p:Prightarrow B$ over an open subset $Vsubseteq B$ and consider the image of $p^{-1}(V)times U$ in $Ptimes_GG/H$, where $U$ here is the same as you consider above.
$endgroup$
– Tyrone
Jan 21 at 9:55
$begingroup$
Thanks Tyrone, I will try spell this out, @Koushik, those are defined as in the notes, we quotient out by the diagonal action when $P times G$ is regarded as a $G$ space, i.e. $(pg,h) sim (p,gh)$. for $p in P, g,h in G$.
$endgroup$
– CL.
Jan 22 at 8:52
add a comment |
$begingroup$
what is your $Ptimes_GG$ and $Ptimes_{G/H}G/H$??
$endgroup$
– Praphulla Koushik
Jan 20 at 16:44
$begingroup$
A principal bundle is trivial if and only if it has a section. Take a trivialisation of $p:Prightarrow B$ over an open subset $Vsubseteq B$ and consider the image of $p^{-1}(V)times U$ in $Ptimes_GG/H$, where $U$ here is the same as you consider above.
$endgroup$
– Tyrone
Jan 21 at 9:55
$begingroup$
Thanks Tyrone, I will try spell this out, @Koushik, those are defined as in the notes, we quotient out by the diagonal action when $P times G$ is regarded as a $G$ space, i.e. $(pg,h) sim (p,gh)$. for $p in P, g,h in G$.
$endgroup$
– CL.
Jan 22 at 8:52
$begingroup$
what is your $Ptimes_GG$ and $Ptimes_{G/H}G/H$??
$endgroup$
– Praphulla Koushik
Jan 20 at 16:44
$begingroup$
what is your $Ptimes_GG$ and $Ptimes_{G/H}G/H$??
$endgroup$
– Praphulla Koushik
Jan 20 at 16:44
$begingroup$
A principal bundle is trivial if and only if it has a section. Take a trivialisation of $p:Prightarrow B$ over an open subset $Vsubseteq B$ and consider the image of $p^{-1}(V)times U$ in $Ptimes_GG/H$, where $U$ here is the same as you consider above.
$endgroup$
– Tyrone
Jan 21 at 9:55
$begingroup$
A principal bundle is trivial if and only if it has a section. Take a trivialisation of $p:Prightarrow B$ over an open subset $Vsubseteq B$ and consider the image of $p^{-1}(V)times U$ in $Ptimes_GG/H$, where $U$ here is the same as you consider above.
$endgroup$
– Tyrone
Jan 21 at 9:55
$begingroup$
Thanks Tyrone, I will try spell this out, @Koushik, those are defined as in the notes, we quotient out by the diagonal action when $P times G$ is regarded as a $G$ space, i.e. $(pg,h) sim (p,gh)$. for $p in P, g,h in G$.
$endgroup$
– CL.
Jan 22 at 8:52
$begingroup$
Thanks Tyrone, I will try spell this out, @Koushik, those are defined as in the notes, we quotient out by the diagonal action when $P times G$ is regarded as a $G$ space, i.e. $(pg,h) sim (p,gh)$. for $p in P, g,h in G$.
$endgroup$
– CL.
Jan 22 at 8:52
add a comment |
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$begingroup$
what is your $Ptimes_GG$ and $Ptimes_{G/H}G/H$??
$endgroup$
– Praphulla Koushik
Jan 20 at 16:44
$begingroup$
A principal bundle is trivial if and only if it has a section. Take a trivialisation of $p:Prightarrow B$ over an open subset $Vsubseteq B$ and consider the image of $p^{-1}(V)times U$ in $Ptimes_GG/H$, where $U$ here is the same as you consider above.
$endgroup$
– Tyrone
Jan 21 at 9:55
$begingroup$
Thanks Tyrone, I will try spell this out, @Koushik, those are defined as in the notes, we quotient out by the diagonal action when $P times G$ is regarded as a $G$ space, i.e. $(pg,h) sim (p,gh)$. for $p in P, g,h in G$.
$endgroup$
– CL.
Jan 22 at 8:52