Equivalence of definitions of a vector bundle
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Let $ninmathbb N$, let $E,B$ be topological spaces and let $p:Eto B$ be a continuous map. For every $bin B$, let $p^-1(b)$ be equipped with the structure of an $n$-dimensional real vector space. Regard the two following definitions of a vector bundle:
$p:Eto B$ is called a vector bundle if for every $bin B$ there is an open neighbourhood $Usubset B$ of $b$ and a homoeomorphism $phi: p^-1(U)to Utimesmathbb R^n$ such that $pi_1circphi=p|_p^-1(U)$, and for all $yin U$ the map $pi_2circphi: p^-1(y)tomathbb R^n$ is linear.
$p:Eto B$ is called a vector bundle if it is a fibre bundle with fibre $mathbb R^n$ and the maps
$$+:Etimes_B Eto E, (x,y)mapsto x+y$$
$$cdot:mathbb Rtimes Eto E, (t,x)mapsto tx$$
are continuous.
Are these definitions equivalent?
If not, are the analogues in the category of smooth manifolds equivalent?
Edit: Removed a superfluous condition in the second definition.
differential-topology vector-bundles fiber-bundles
asked Sep 10 at 19:29
jb78685
445
add a comment |Â
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Let $ninmathbb N$, let $E,B$ be topological spaces and let $p:Eto B$ be a continuous map. For every $bin B$, let $p^-1(b)$ be equipped with the structure of an $n$-dimensional real vector space. Regard the two following definitions of a vector bundle:
$p:Eto B$ is called a vector bundle if for every $bin B$ there is an open neighbourhood $Usubset B$ of $b$ and a homoeomorphism $phi: p^-1(U)to Utimesmathbb R^n$ such that $pi_1circphi=p|_p^-1(U)$, and for all $yin U$ the map $pi_2circphi: p^-1(y)tomathbb R^n$ is linear.
$p:Eto B$ is called a vector bundle if it is a fibre bundle with fibre $mathbb R^n$ and the maps
$$+:Etimes_B Eto E, (x,y)mapsto x+y$$
$$cdot:mathbb Rtimes Eto E, (t,x)mapsto tx$$
are continuous.
Are these definitions equivalent?
If not, are the analogues in the category of smooth manifolds equivalent?
Edit: Removed a superfluous condition in the second definition.
differential-topology vector-bundles fiber-bundles
asked Sep 10 at 19:29
jb78685
445
Where did you find definition 2?
– md2perpe
Sep 12 at 12:54
I did not find the second definition anywhere. It seemed natural to me that, if a fibre bundle has these properties, its local trivialisations could be tweaked to be fibre-wise linear, at least in the smooth case. I cannot quite show that the definitions are equivalent and I actually assume that they are not, but I cannot think of a counter example.
– jb78685
Sep 12 at 13:25
The definitions of $+$ and $cdot$ are not clear to me. Are they defined pointwise over $B$ so that $e_1 + e_2$ is defined if $p(e_1) = p(e_2) =: b$ and then $e_1 + e_2 = phi^-1(b, pi_2circphi(e_1) + pi_2circphi(e_2))$?
– md2perpe
Sep 12 at 14:14
$Etimes_B E=p(e_1)=p(e_2)$ is the fibre product of $E$ with itself with respect to $p$. For every $bin B$, $p^-1(b)$ is already an $n$-dimensional real vector space. The maps $+$ and $cdot$ are defined through these vector space operations. A priori, a vector bundle according to definition 2 need not admit local trivialisations that are linear in each fibre.
– jb78685
Sep 12 at 15:43
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $ninmathbb N$, let $E,B$ be topological spaces and let $p:Eto B$ be a continuous map. For every $bin B$, let $p^-1(b)$ be equipped with the structure of an $n$-dimensional real vector space. Regard the two following definitions of a vector bundle:
$p:Eto B$ is called a vector bundle if for every $bin B$ there is an open neighbourhood $Usubset B$ of $b$ and a homoeomorphism $phi: p^-1(U)to Utimesmathbb R^n$ such that $pi_1circphi=p|_p^-1(U)$, and for all $yin U$ the map $pi_2circphi: p^-1(y)tomathbb R^n$ is linear.
$p:Eto B$ is called a vector bundle if it is a fibre bundle with fibre $mathbb R^n$ and the maps
$$+:Etimes_B Eto E, (x,y)mapsto x+y$$
$$cdot:mathbb Rtimes Eto E, (t,x)mapsto tx$$
are continuous.
Are these definitions equivalent?
If not, are the analogues in the category of smooth manifolds equivalent?
Edit: Removed a superfluous condition in the second definition.
differential-topology vector-bundles fiber-bundles
asked Sep 10 at 19:29
jb78685
445
Let $ninmathbb N$, let $E,B$ be topological spaces and let $p:Eto B$ be a continuous map. For every $bin B$, let $p^-1(b)$ be equipped with the structure of an $n$-dimensional real vector space. Regard the two following definitions of a vector bundle:
$p:Eto B$ is called a vector bundle if for every $bin B$ there is an open neighbourhood $Usubset B$ of $b$ and a homoeomorphism $phi: p^-1(U)to Utimesmathbb R^n$ such that $pi_1circphi=p|_p^-1(U)$, and for all $yin U$ the map $pi_2circphi: p^-1(y)tomathbb R^n$ is linear.
$p:Eto B$ is called a vector bundle if it is a fibre bundle with fibre $mathbb R^n$ and the maps
$$+:Etimes_B Eto E, (x,y)mapsto x+y$$
$$cdot:mathbb Rtimes Eto E, (t,x)mapsto tx$$
are continuous.
Are these definitions equivalent?
If not, are the analogues in the category of smooth manifolds equivalent?
Edit: Removed a superfluous condition in the second definition.
differential-topology vector-bundles fiber-bundles
differential-topology vector-bundles fiber-bundles
asked Sep 10 at 19:29
jb78685
445
asked Sep 10 at 19:29
jb78685
445
edited Sep 13 at 9:34
asked Sep 10 at 19:29
jb78685
445
asked Sep 10 at 19:29
jb78685
445
asked Sep 10 at 19:29
jb78685
445
445
Where did you find definition 2?
– md2perpe
Sep 12 at 12:54
I did not find the second definition anywhere. It seemed natural to me that, if a fibre bundle has these properties, its local trivialisations could be tweaked to be fibre-wise linear, at least in the smooth case. I cannot quite show that the definitions are equivalent and I actually assume that they are not, but I cannot think of a counter example.
– jb78685
Sep 12 at 13:25
The definitions of $+$ and $cdot$ are not clear to me. Are they defined pointwise over $B$ so that $e_1 + e_2$ is defined if $p(e_1) = p(e_2) =: b$ and then $e_1 + e_2 = phi^-1(b, pi_2circphi(e_1) + pi_2circphi(e_2))$?
– md2perpe
Sep 12 at 14:14
$Etimes_B E=p(e_1)=p(e_2)$ is the fibre product of $E$ with itself with respect to $p$. For every $bin B$, $p^-1(b)$ is already an $n$-dimensional real vector space. The maps $+$ and $cdot$ are defined through these vector space operations. A priori, a vector bundle according to definition 2 need not admit local trivialisations that are linear in each fibre.
– jb78685
Sep 12 at 15:43
add a comment |Â
Where did you find definition 2?
– md2perpe
Sep 12 at 12:54
I did not find the second definition anywhere. It seemed natural to me that, if a fibre bundle has these properties, its local trivialisations could be tweaked to be fibre-wise linear, at least in the smooth case. I cannot quite show that the definitions are equivalent and I actually assume that they are not, but I cannot think of a counter example.
– jb78685
Sep 12 at 13:25
The definitions of $+$ and $cdot$ are not clear to me. Are they defined pointwise over $B$ so that $e_1 + e_2$ is defined if $p(e_1) = p(e_2) =: b$ and then $e_1 + e_2 = phi^-1(b, pi_2circphi(e_1) + pi_2circphi(e_2))$?
– md2perpe
Sep 12 at 14:14
$Etimes_B E=p(e_1)=p(e_2)$ is the fibre product of $E$ with itself with respect to $p$. For every $bin B$, $p^-1(b)$ is already an $n$-dimensional real vector space. The maps $+$ and $cdot$ are defined through these vector space operations. A priori, a vector bundle according to definition 2 need not admit local trivialisations that are linear in each fibre.
– jb78685
Sep 12 at 15:43
Where did you find definition 2?
– md2perpe
Sep 12 at 12:54
Where did you find definition 2?
– md2perpe
Sep 12 at 12:54
I did not find the second definition anywhere. It seemed natural to me that, if a fibre bundle has these properties, its local trivialisations could be tweaked to be fibre-wise linear, at least in the smooth case. I cannot quite show that the definitions are equivalent and I actually assume that they are not, but I cannot think of a counter example.
– jb78685
Sep 12 at 13:25
I did not find the second definition anywhere. It seemed natural to me that, if a fibre bundle has these properties, its local trivialisations could be tweaked to be fibre-wise linear, at least in the smooth case. I cannot quite show that the definitions are equivalent and I actually assume that they are not, but I cannot think of a counter example.
– jb78685
Sep 12 at 13:25
The definitions of $+$ and $cdot$ are not clear to me. Are they defined pointwise over $B$ so that $e_1 + e_2$ is defined if $p(e_1) = p(e_2) =: b$ and then $e_1 + e_2 = phi^-1(b, pi_2circphi(e_1) + pi_2circphi(e_2))$?
– md2perpe
Sep 12 at 14:14
The definitions of $+$ and $cdot$ are not clear to me. Are they defined pointwise over $B$ so that $e_1 + e_2$ is defined if $p(e_1) = p(e_2) =: b$ and then $e_1 + e_2 = phi^-1(b, pi_2circphi(e_1) + pi_2circphi(e_2))$?
– md2perpe
Sep 12 at 14:14
$Etimes_B E=p(e_1)=p(e_2)$ is the fibre product of $E$ with itself with respect to $p$. For every $bin B$, $p^-1(b)$ is already an $n$-dimensional real vector space. The maps $+$ and $cdot$ are defined through these vector space operations. A priori, a vector bundle according to definition 2 need not admit local trivialisations that are linear in each fibre.
– jb78685
Sep 12 at 15:43
$Etimes_B E=p(e_1)=p(e_2)$ is the fibre product of $E$ with itself with respect to $p$. For every $bin B$, $p^-1(b)$ is already an $n$-dimensional real vector space. The maps $+$ and $cdot$ are defined through these vector space operations. A priori, a vector bundle according to definition 2 need not admit local trivialisations that are linear in each fibre.
– jb78685
Sep 12 at 15:43
add a comment |Â
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Where did you find definition 2?
– md2perpe
Sep 12 at 12:54
I did not find the second definition anywhere. It seemed natural to me that, if a fibre bundle has these properties, its local trivialisations could be tweaked to be fibre-wise linear, at least in the smooth case. I cannot quite show that the definitions are equivalent and I actually assume that they are not, but I cannot think of a counter example.
– jb78685
Sep 12 at 13:25
The definitions of $+$ and $cdot$ are not clear to me. Are they defined pointwise over $B$ so that $e_1 + e_2$ is defined if $p(e_1) = p(e_2) =: b$ and then $e_1 + e_2 = phi^-1(b, pi_2circphi(e_1) + pi_2circphi(e_2))$?
– md2perpe
Sep 12 at 14:14
$Etimes_B E=p(e_1)=p(e_2)$ is the fibre product of $E$ with itself with respect to $p$. For every $bin B$, $p^-1(b)$ is already an $n$-dimensional real vector space. The maps $+$ and $cdot$ are defined through these vector space operations. A priori, a vector bundle according to definition 2 need not admit local trivialisations that are linear in each fibre.
– jb78685
Sep 12 at 15:43