Mixing
Bundle of Endomorphism of Line Bundle always trivial
$begingroup$
Let $B$ a topologiocal space (I'm not sure if it should be nice enough for the following statement ... eg paracompact).
Let $L$ be a line bundle over $B$.
My question is how to see that the morphism bundle $underline{Hom}(L,L)$ is a trivial bundle? (here $underline{Hom}(L,L)$ means that it is fiberwise $b$ the space linear maps between one dimensional spaces $L_b$)
My attempts: Obvioulsy it suffice to show that there exist a non vanishing glocal section $B to underline{Hom}(L,L)$.
Intuitively I would guess that the map $b to id_{L_b}in Hom(L_b,L_b)$ should work. Is it ok or is it a bit more complicated and I have overseen a subtle detail?
general-topology vector-bundles line-bundles
asked Jan 25 at 18:25
KarlPeterKarlPeter
4641315
$endgroup$
add a comment |
$begingroup$
Let $B$ a topologiocal space (I'm not sure if it should be nice enough for the following statement ... eg paracompact).
Let $L$ be a line bundle over $B$.
My question is how to see that the morphism bundle $underline{Hom}(L,L)$ is a trivial bundle? (here $underline{Hom}(L,L)$ means that it is fiberwise $b$ the space linear maps between one dimensional spaces $L_b$)
My attempts: Obvioulsy it suffice to show that there exist a non vanishing glocal section $B to underline{Hom}(L,L)$.
Intuitively I would guess that the map $b to id_{L_b}in Hom(L_b,L_b)$ should work. Is it ok or is it a bit more complicated and I have overseen a subtle detail?
general-topology vector-bundles line-bundles
asked Jan 25 at 18:25
KarlPeterKarlPeter
4641315
$endgroup$
$begingroup$
Your proof is fine; the identity maps of the fibers constitute a nowhere-vanishing section of the bundle of endomorphisms. (But the title of your question is false; you need that the morphisms are between a line bundle and itself, not some other line bundle.)
$endgroup$
– Andreas Blass
Jan 25 at 18:35
add a comment |
$begingroup$
Let $B$ a topologiocal space (I'm not sure if it should be nice enough for the following statement ... eg paracompact).
Let $L$ be a line bundle over $B$.
My question is how to see that the morphism bundle $underline{Hom}(L,L)$ is a trivial bundle? (here $underline{Hom}(L,L)$ means that it is fiberwise $b$ the space linear maps between one dimensional spaces $L_b$)
My attempts: Obvioulsy it suffice to show that there exist a non vanishing glocal section $B to underline{Hom}(L,L)$.
Intuitively I would guess that the map $b to id_{L_b}in Hom(L_b,L_b)$ should work. Is it ok or is it a bit more complicated and I have overseen a subtle detail?
general-topology vector-bundles line-bundles
asked Jan 25 at 18:25
KarlPeterKarlPeter
4641315
$endgroup$
Let $B$ a topologiocal space (I'm not sure if it should be nice enough for the following statement ... eg paracompact).
Let $L$ be a line bundle over $B$.
My question is how to see that the morphism bundle $underline{Hom}(L,L)$ is a trivial bundle? (here $underline{Hom}(L,L)$ means that it is fiberwise $b$ the space linear maps between one dimensional spaces $L_b$)
My attempts: Obvioulsy it suffice to show that there exist a non vanishing glocal section $B to underline{Hom}(L,L)$.
Intuitively I would guess that the map $b to id_{L_b}in Hom(L_b,L_b)$ should work. Is it ok or is it a bit more complicated and I have overseen a subtle detail?
general-topology vector-bundles line-bundles
general-topology vector-bundles line-bundles
asked Jan 25 at 18:25
KarlPeterKarlPeter
4641315
asked Jan 25 at 18:25
KarlPeterKarlPeter
4641315
edited Jan 25 at 18:43
KarlPeter
asked Jan 25 at 18:25
KarlPeterKarlPeter
4641315
asked Jan 25 at 18:25
KarlPeterKarlPeter
4641315
asked Jan 25 at 18:25
KarlPeterKarlPeter
4641315
4641315
$begingroup$
Your proof is fine; the identity maps of the fibers constitute a nowhere-vanishing section of the bundle of endomorphisms. (But the title of your question is false; you need that the morphisms are between a line bundle and itself, not some other line bundle.)
$endgroup$
– Andreas Blass
Jan 25 at 18:35
add a comment |
$begingroup$
Your proof is fine; the identity maps of the fibers constitute a nowhere-vanishing section of the bundle of endomorphisms. (But the title of your question is false; you need that the morphisms are between a line bundle and itself, not some other line bundle.)
$endgroup$
– Andreas Blass
Jan 25 at 18:35
$begingroup$
Your proof is fine; the identity maps of the fibers constitute a nowhere-vanishing section of the bundle of endomorphisms. (But the title of your question is false; you need that the morphisms are between a line bundle and itself, not some other line bundle.)
$endgroup$
– Andreas Blass
Jan 25 at 18:35
$begingroup$
Your proof is fine; the identity maps of the fibers constitute a nowhere-vanishing section of the bundle of endomorphisms. (But the title of your question is false; you need that the morphisms are between a line bundle and itself, not some other line bundle.)
$endgroup$
– Andreas Blass
Jan 25 at 18:35
add a comment |
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$begingroup$
Your proof is fine; the identity maps of the fibers constitute a nowhere-vanishing section of the bundle of endomorphisms. (But the title of your question is false; you need that the morphisms are between a line bundle and itself, not some other line bundle.)
$endgroup$
– Andreas Blass
Jan 25 at 18:35