Mixing
An isomorphism of vector bundles over a manifold, $K(X)$,
$begingroup$
Let $E_1, E_0$ be vector bundles over a manifold $X$. Let us suppose that
$$E_1 - E_0 =0 in K(X)$$
(I believe we also suppose $X$ to be compact so $K$-theory makes sense here.)
Proposition: If $dim E_i > dim X$, then there is an isomorphism,
$$ E_1 rightarrow E_0$$
over $X$.
I am greatful for references or proof. I have not found any material regarding bundles over manifolds.
Source: This is claimed in page 20 line 12 of this paper by Atiyah.
reference-request algebraic-topology differential-topology vector-bundles k-theory
asked Jan 31 at 17:09
CL.CL.
2,3102925
$endgroup$
add a comment |
$begingroup$
Let $E_1, E_0$ be vector bundles over a manifold $X$. Let us suppose that
$$E_1 - E_0 =0 in K(X)$$
(I believe we also suppose $X$ to be compact so $K$-theory makes sense here.)
Proposition: If $dim E_i > dim X$, then there is an isomorphism,
$$ E_1 rightarrow E_0$$
over $X$.
I am greatful for references or proof. I have not found any material regarding bundles over manifolds.
Source: This is claimed in page 20 line 12 of this paper by Atiyah.
reference-request algebraic-topology differential-topology vector-bundles k-theory
asked Jan 31 at 17:09
CL.CL.
2,3102925
$endgroup$
$begingroup$
Your link is broken.
$endgroup$
– user98602
Jan 31 at 17:10
$begingroup$
Thanks a lot, corrected.
$endgroup$
– CL.
Jan 31 at 17:21
add a comment |
$begingroup$
Let $E_1, E_0$ be vector bundles over a manifold $X$. Let us suppose that
$$E_1 - E_0 =0 in K(X)$$
(I believe we also suppose $X$ to be compact so $K$-theory makes sense here.)
Proposition: If $dim E_i > dim X$, then there is an isomorphism,
$$ E_1 rightarrow E_0$$
over $X$.
I am greatful for references or proof. I have not found any material regarding bundles over manifolds.
Source: This is claimed in page 20 line 12 of this paper by Atiyah.
reference-request algebraic-topology differential-topology vector-bundles k-theory
asked Jan 31 at 17:09
CL.CL.
2,3102925
$endgroup$
Let $E_1, E_0$ be vector bundles over a manifold $X$. Let us suppose that
$$E_1 - E_0 =0 in K(X)$$
(I believe we also suppose $X$ to be compact so $K$-theory makes sense here.)
Proposition: If $dim E_i > dim X$, then there is an isomorphism,
$$ E_1 rightarrow E_0$$
over $X$.
I am greatful for references or proof. I have not found any material regarding bundles over manifolds.
Source: This is claimed in page 20 line 12 of this paper by Atiyah.
reference-request algebraic-topology differential-topology vector-bundles k-theory
reference-request algebraic-topology differential-topology vector-bundles k-theory
asked Jan 31 at 17:09
CL.CL.
2,3102925
asked Jan 31 at 17:09
CL.CL.
2,3102925
edited Jan 31 at 17:10
CL.
asked Jan 31 at 17:09
CL.CL.
2,3102925
asked Jan 31 at 17:09
CL.CL.
2,3102925
asked Jan 31 at 17:09
CL.CL.
2,3102925
2,3102925
$begingroup$
Your link is broken.
$endgroup$
– user98602
Jan 31 at 17:10
$begingroup$
Thanks a lot, corrected.
$endgroup$
– CL.
Jan 31 at 17:21
add a comment |
$begingroup$
Your link is broken.
$endgroup$
– user98602
Jan 31 at 17:10
$begingroup$
Thanks a lot, corrected.
$endgroup$
– CL.
Jan 31 at 17:21
$begingroup$
Your link is broken.
$endgroup$
– user98602
Jan 31 at 17:10
$begingroup$
Your link is broken.
$endgroup$
– user98602
Jan 31 at 17:10
$begingroup$
Thanks a lot, corrected.
$endgroup$
– CL.
Jan 31 at 17:21
$begingroup$
Thanks a lot, corrected.
$endgroup$
– CL.
Jan 31 at 17:21
add a comment |
1 Answer
1
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$begingroup$
This follows from the existence of classifying spaces. First let me work with reduced K-theory, as if $E_1 = E_2$ in $K$-theory itself, then the rank of $E_1$ and $E_2$ are the same, and conversely if $E_1$ and $E_2$ have the same rank and they are equal in reduced $K$-theory, then they are equal in $K$-theory.
We have $$tilde K^0(X) = [X, BO],$$ while $$text{Vect}_n(X) = [X, BO(n)].$$
The map $text{Vect}_n(X) to tilde K^0(X)$ is given by sending $BO(n) hookrightarrow BO$.
Lemma: if $n > dim X$, the map $text{Vect}_n(X) to text{Vect}_{n+1}(X)$, given by sending $BO(n) to BO(n+1)$, is a bijection.
Proof: there is a fibration $S^n to BO(n) to BO(n+1)$, which induces an exact sequence on mapping sets $[X, S^n] to [X, BO(n)] to [X, BO(n+1)]$; because $[X, S^n] = 0$ as $n > dim X$, the map $text{Vect}_n(X) to text{Vect}_{n+1}(X)$ is injective. For surjectivity (which applies one dimension lower), we need to show that any bundle of rank $n > dim X$ has a trivial line bundle as summand. Equivalently, we need to show that every sphere bundle $S^{n-1} hookrightarrow E to X$ has a section as soon as $n > dim X$. This follows from obstruction theory, which identifies the obstruction to the existence of a section as an element of the set $H^n(X;pi_{n-1} S^{n-1}) = 0$, by the assumption that $n > dim X$.
Because $BO = text{colim }BO(n)$, you have (so long as your spaces aren't silly) an equality $$tilde K^0(X) = [X, BO] = text{colim } [X, BO(n)] = text{colim Vect}_n(X).$$
By the lemma, so long as $n > dim X$, the map $text{Vect}_n(X) to tilde K^0(X)$ is a bijection. This is what you wanted.
$endgroup$
$begingroup$
Thanks a lot for such a detailed reply, it would probably take me some time to digest this - what is a good elementary reference for some background in classifying spaces?
$endgroup$
– CL.
Jan 31 at 17:48
$begingroup$
The canonical reference is Milnor and Stasheff's book on characteristic classes. Hatcher has some discussion in his vector bundle notes.
$endgroup$
– user98602
Jan 31 at 17:51
add a comment |
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$begingroup$
This follows from the existence of classifying spaces. First let me work with reduced K-theory, as if $E_1 = E_2$ in $K$-theory itself, then the rank of $E_1$ and $E_2$ are the same, and conversely if $E_1$ and $E_2$ have the same rank and they are equal in reduced $K$-theory, then they are equal in $K$-theory.
We have $$tilde K^0(X) = [X, BO],$$ while $$text{Vect}_n(X) = [X, BO(n)].$$
The map $text{Vect}_n(X) to tilde K^0(X)$ is given by sending $BO(n) hookrightarrow BO$.
Lemma: if $n > dim X$, the map $text{Vect}_n(X) to text{Vect}_{n+1}(X)$, given by sending $BO(n) to BO(n+1)$, is a bijection.
Proof: there is a fibration $S^n to BO(n) to BO(n+1)$, which induces an exact sequence on mapping sets $[X, S^n] to [X, BO(n)] to [X, BO(n+1)]$; because $[X, S^n] = 0$ as $n > dim X$, the map $text{Vect}_n(X) to text{Vect}_{n+1}(X)$ is injective. For surjectivity (which applies one dimension lower), we need to show that any bundle of rank $n > dim X$ has a trivial line bundle as summand. Equivalently, we need to show that every sphere bundle $S^{n-1} hookrightarrow E to X$ has a section as soon as $n > dim X$. This follows from obstruction theory, which identifies the obstruction to the existence of a section as an element of the set $H^n(X;pi_{n-1} S^{n-1}) = 0$, by the assumption that $n > dim X$.
Because $BO = text{colim }BO(n)$, you have (so long as your spaces aren't silly) an equality $$tilde K^0(X) = [X, BO] = text{colim } [X, BO(n)] = text{colim Vect}_n(X).$$
By the lemma, so long as $n > dim X$, the map $text{Vect}_n(X) to tilde K^0(X)$ is a bijection. This is what you wanted.
$endgroup$
$begingroup$
Thanks a lot for such a detailed reply, it would probably take me some time to digest this - what is a good elementary reference for some background in classifying spaces?
$endgroup$
– CL.
Jan 31 at 17:48
$begingroup$
The canonical reference is Milnor and Stasheff's book on characteristic classes. Hatcher has some discussion in his vector bundle notes.
$endgroup$
– user98602
Jan 31 at 17:51
add a comment |
$begingroup$
This follows from the existence of classifying spaces. First let me work with reduced K-theory, as if $E_1 = E_2$ in $K$-theory itself, then the rank of $E_1$ and $E_2$ are the same, and conversely if $E_1$ and $E_2$ have the same rank and they are equal in reduced $K$-theory, then they are equal in $K$-theory.
We have $$tilde K^0(X) = [X, BO],$$ while $$text{Vect}_n(X) = [X, BO(n)].$$
The map $text{Vect}_n(X) to tilde K^0(X)$ is given by sending $BO(n) hookrightarrow BO$.
Lemma: if $n > dim X$, the map $text{Vect}_n(X) to text{Vect}_{n+1}(X)$, given by sending $BO(n) to BO(n+1)$, is a bijection.
Proof: there is a fibration $S^n to BO(n) to BO(n+1)$, which induces an exact sequence on mapping sets $[X, S^n] to [X, BO(n)] to [X, BO(n+1)]$; because $[X, S^n] = 0$ as $n > dim X$, the map $text{Vect}_n(X) to text{Vect}_{n+1}(X)$ is injective. For surjectivity (which applies one dimension lower), we need to show that any bundle of rank $n > dim X$ has a trivial line bundle as summand. Equivalently, we need to show that every sphere bundle $S^{n-1} hookrightarrow E to X$ has a section as soon as $n > dim X$. This follows from obstruction theory, which identifies the obstruction to the existence of a section as an element of the set $H^n(X;pi_{n-1} S^{n-1}) = 0$, by the assumption that $n > dim X$.
Because $BO = text{colim }BO(n)$, you have (so long as your spaces aren't silly) an equality $$tilde K^0(X) = [X, BO] = text{colim } [X, BO(n)] = text{colim Vect}_n(X).$$
By the lemma, so long as $n > dim X$, the map $text{Vect}_n(X) to tilde K^0(X)$ is a bijection. This is what you wanted.
$endgroup$
$begingroup$
Thanks a lot for such a detailed reply, it would probably take me some time to digest this - what is a good elementary reference for some background in classifying spaces?
$endgroup$
– CL.
Jan 31 at 17:48
$begingroup$
The canonical reference is Milnor and Stasheff's book on characteristic classes. Hatcher has some discussion in his vector bundle notes.
$endgroup$
– user98602
Jan 31 at 17:51
add a comment |
$begingroup$
This follows from the existence of classifying spaces. First let me work with reduced K-theory, as if $E_1 = E_2$ in $K$-theory itself, then the rank of $E_1$ and $E_2$ are the same, and conversely if $E_1$ and $E_2$ have the same rank and they are equal in reduced $K$-theory, then they are equal in $K$-theory.
We have $$tilde K^0(X) = [X, BO],$$ while $$text{Vect}_n(X) = [X, BO(n)].$$
The map $text{Vect}_n(X) to tilde K^0(X)$ is given by sending $BO(n) hookrightarrow BO$.
Lemma: if $n > dim X$, the map $text{Vect}_n(X) to text{Vect}_{n+1}(X)$, given by sending $BO(n) to BO(n+1)$, is a bijection.
Proof: there is a fibration $S^n to BO(n) to BO(n+1)$, which induces an exact sequence on mapping sets $[X, S^n] to [X, BO(n)] to [X, BO(n+1)]$; because $[X, S^n] = 0$ as $n > dim X$, the map $text{Vect}_n(X) to text{Vect}_{n+1}(X)$ is injective. For surjectivity (which applies one dimension lower), we need to show that any bundle of rank $n > dim X$ has a trivial line bundle as summand. Equivalently, we need to show that every sphere bundle $S^{n-1} hookrightarrow E to X$ has a section as soon as $n > dim X$. This follows from obstruction theory, which identifies the obstruction to the existence of a section as an element of the set $H^n(X;pi_{n-1} S^{n-1}) = 0$, by the assumption that $n > dim X$.
Because $BO = text{colim }BO(n)$, you have (so long as your spaces aren't silly) an equality $$tilde K^0(X) = [X, BO] = text{colim } [X, BO(n)] = text{colim Vect}_n(X).$$
By the lemma, so long as $n > dim X$, the map $text{Vect}_n(X) to tilde K^0(X)$ is a bijection. This is what you wanted.
$endgroup$
This follows from the existence of classifying spaces. First let me work with reduced K-theory, as if $E_1 = E_2$ in $K$-theory itself, then the rank of $E_1$ and $E_2$ are the same, and conversely if $E_1$ and $E_2$ have the same rank and they are equal in reduced $K$-theory, then they are equal in $K$-theory.
We have $$tilde K^0(X) = [X, BO],$$ while $$text{Vect}_n(X) = [X, BO(n)].$$
The map $text{Vect}_n(X) to tilde K^0(X)$ is given by sending $BO(n) hookrightarrow BO$.
Lemma: if $n > dim X$, the map $text{Vect}_n(X) to text{Vect}_{n+1}(X)$, given by sending $BO(n) to BO(n+1)$, is a bijection.
Proof: there is a fibration $S^n to BO(n) to BO(n+1)$, which induces an exact sequence on mapping sets $[X, S^n] to [X, BO(n)] to [X, BO(n+1)]$; because $[X, S^n] = 0$ as $n > dim X$, the map $text{Vect}_n(X) to text{Vect}_{n+1}(X)$ is injective. For surjectivity (which applies one dimension lower), we need to show that any bundle of rank $n > dim X$ has a trivial line bundle as summand. Equivalently, we need to show that every sphere bundle $S^{n-1} hookrightarrow E to X$ has a section as soon as $n > dim X$. This follows from obstruction theory, which identifies the obstruction to the existence of a section as an element of the set $H^n(X;pi_{n-1} S^{n-1}) = 0$, by the assumption that $n > dim X$.
Because $BO = text{colim }BO(n)$, you have (so long as your spaces aren't silly) an equality $$tilde K^0(X) = [X, BO] = text{colim } [X, BO(n)] = text{colim Vect}_n(X).$$
By the lemma, so long as $n > dim X$, the map $text{Vect}_n(X) to tilde K^0(X)$ is a bijection. This is what you wanted.
answered Jan 31 at 17:28
user98602
$begingroup$
Thanks a lot for such a detailed reply, it would probably take me some time to digest this - what is a good elementary reference for some background in classifying spaces?
$endgroup$
– CL.
Jan 31 at 17:48
$begingroup$
The canonical reference is Milnor and Stasheff's book on characteristic classes. Hatcher has some discussion in his vector bundle notes.
$endgroup$
– user98602
Jan 31 at 17:51
add a comment |
$begingroup$
Thanks a lot for such a detailed reply, it would probably take me some time to digest this - what is a good elementary reference for some background in classifying spaces?
$endgroup$
– CL.
Jan 31 at 17:48
$begingroup$
The canonical reference is Milnor and Stasheff's book on characteristic classes. Hatcher has some discussion in his vector bundle notes.
$endgroup$
– user98602
Jan 31 at 17:51
$begingroup$
Thanks a lot for such a detailed reply, it would probably take me some time to digest this - what is a good elementary reference for some background in classifying spaces?
$endgroup$
– CL.
Jan 31 at 17:48
$begingroup$
Thanks a lot for such a detailed reply, it would probably take me some time to digest this - what is a good elementary reference for some background in classifying spaces?
$endgroup$
– CL.
Jan 31 at 17:48
$begingroup$
The canonical reference is Milnor and Stasheff's book on characteristic classes. Hatcher has some discussion in his vector bundle notes.
$endgroup$
– user98602
Jan 31 at 17:51
$begingroup$
The canonical reference is Milnor and Stasheff's book on characteristic classes. Hatcher has some discussion in his vector bundle notes.
$endgroup$
– user98602
Jan 31 at 17:51
add a comment |
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$begingroup$
Your link is broken.
$endgroup$
– user98602
Jan 31 at 17:10
$begingroup$
Thanks a lot, corrected.
$endgroup$
– CL.
Jan 31 at 17:21