Mixing
Isomorphism of Principal Bundles with structure groupoid
$begingroup$
Let $mathcal{G}rightrightarrows M$ be a Lie groupoid. Suppose that $pi:Prightarrow B$ is a $mathcal{G}$-principal bundle and let $(h_s,g_s):(P,B)rightarrow (P,B); sin [0,1]$, be a homotopy of $mathcal{G}$-principal bundle morphisms, that is:
for each $sin [0,1]$ we have $h_s:Prightarrow P$, $mathcal{G}$-equivariant map and $g_s:Brightarrow B$, such that $picirc h_s=g_scirc pi$, in such a way that the families $(h_s)_{sin [0,1]}$ and $(g_s)_{sin [0,1]}$ are homotopies.
I want to prove that the $mathcal{G}$-bundles $g_0^{ast}(P)rightarrow B$ and $g_1^{ast}(P)rightarrow B$ are isomorphic to each other. Is that true? Can anyone give a proof or a reference please?
lie-groups principal-bundles groupoids
asked Mar 26 '15 at 3:30
StudzinskiStudzinski
9741518
$endgroup$
add a comment |
$begingroup$
Let $mathcal{G}rightrightarrows M$ be a Lie groupoid. Suppose that $pi:Prightarrow B$ is a $mathcal{G}$-principal bundle and let $(h_s,g_s):(P,B)rightarrow (P,B); sin [0,1]$, be a homotopy of $mathcal{G}$-principal bundle morphisms, that is:
for each $sin [0,1]$ we have $h_s:Prightarrow P$, $mathcal{G}$-equivariant map and $g_s:Brightarrow B$, such that $picirc h_s=g_scirc pi$, in such a way that the families $(h_s)_{sin [0,1]}$ and $(g_s)_{sin [0,1]}$ are homotopies.
I want to prove that the $mathcal{G}$-bundles $g_0^{ast}(P)rightarrow B$ and $g_1^{ast}(P)rightarrow B$ are isomorphic to each other. Is that true? Can anyone give a proof or a reference please?
lie-groups principal-bundles groupoids
asked Mar 26 '15 at 3:30
StudzinskiStudzinski
9741518
$endgroup$
add a comment |
$begingroup$
Let $mathcal{G}rightrightarrows M$ be a Lie groupoid. Suppose that $pi:Prightarrow B$ is a $mathcal{G}$-principal bundle and let $(h_s,g_s):(P,B)rightarrow (P,B); sin [0,1]$, be a homotopy of $mathcal{G}$-principal bundle morphisms, that is:
for each $sin [0,1]$ we have $h_s:Prightarrow P$, $mathcal{G}$-equivariant map and $g_s:Brightarrow B$, such that $picirc h_s=g_scirc pi$, in such a way that the families $(h_s)_{sin [0,1]}$ and $(g_s)_{sin [0,1]}$ are homotopies.
I want to prove that the $mathcal{G}$-bundles $g_0^{ast}(P)rightarrow B$ and $g_1^{ast}(P)rightarrow B$ are isomorphic to each other. Is that true? Can anyone give a proof or a reference please?
lie-groups principal-bundles groupoids
asked Mar 26 '15 at 3:30
StudzinskiStudzinski
9741518
$endgroup$
Let $mathcal{G}rightrightarrows M$ be a Lie groupoid. Suppose that $pi:Prightarrow B$ is a $mathcal{G}$-principal bundle and let $(h_s,g_s):(P,B)rightarrow (P,B); sin [0,1]$, be a homotopy of $mathcal{G}$-principal bundle morphisms, that is:
for each $sin [0,1]$ we have $h_s:Prightarrow P$, $mathcal{G}$-equivariant map and $g_s:Brightarrow B$, such that $picirc h_s=g_scirc pi$, in such a way that the families $(h_s)_{sin [0,1]}$ and $(g_s)_{sin [0,1]}$ are homotopies.
I want to prove that the $mathcal{G}$-bundles $g_0^{ast}(P)rightarrow B$ and $g_1^{ast}(P)rightarrow B$ are isomorphic to each other. Is that true? Can anyone give a proof or a reference please?
lie-groups principal-bundles groupoids
lie-groups principal-bundles groupoids
asked Mar 26 '15 at 3:30
StudzinskiStudzinski
9741518
asked Mar 26 '15 at 3:30
StudzinskiStudzinski
9741518
asked Mar 26 '15 at 3:30
StudzinskiStudzinski
9741518
asked Mar 26 '15 at 3:30
StudzinskiStudzinski
9741518
asked Mar 26 '15 at 3:30
StudzinskiStudzinski
9741518
9741518
add a comment |
add a comment |
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