Chern class of a principal $G$ bundle for a compact Lie group $G$
This question is related to this question. The user who asked this question is not active since September. So, asking a separate question here.
Let $G$ be a compact Lie group and $Prightarrow M$ be a principal $G$ bundle. We want to associate Chern classes for this bundle.
This article says in its second page that "A compact Lie group $G$ may be shown (with some work, using the theory of compact operators) to be none other than a closed subgroup of the Unitary group $U(n)$ of matrices
for some $n$". Thus, for structure group $G$ of $P(M,G)$ there exists an embedding $Ghookrightarrow U(n)$.
See the embedding $Ghookrightarrow U(n)$ as an action of $G$ on $U(n)$. Given a manifold $P'$ and an action of $G$ on $P'$ there is a notion of what is called an associated bundle whose fibres are $P'$. In case $P'$ is a Lie group $H$, we get a principal $H$ bundle. This $G$ action on $U(n)$ gives a principal $U(n)$ bundle $Qrightarrow M$.
For this principal $U(n)$ bundle $Qrightarrow M$, we fix a connection $Gamma$ with curvature $Omega$. We then get what is called Weil homomorphism $W:I(U(n))rightarrow H^*(M,mathbb{R})$.
We choose elements $f_k:mathfrak{u}(n,mathbb{C})rightarrow mathbb{R}$ from $I(U(n))$ such that $$text{det}left(lambda I-frac{1}{2pisqrt{-1}}Xright)=sum_{k=0}^n f_k(X)lambda^{n-k}$$
The images $W(f_k)in H^{2k}(M,mathbb{R})$ are called the $k$-th Chern classes of $Qrightarrow M$.
How do we define Chern classes of the principal $G$ bundle $Prightarrow M$ that we have started with?
Is the $k$-th Chern class of $G$ bundle $P(M,G)$ the $k$-th Chern class of the $U(n)$ bundle $Q(M,U(n))$?
Does this depend on the embedding $Ghookrightarrow U(n)$ we have chosen?
In case it is dependent on the choice of $Ghookrightarrow U(n)$, what invariants of $P(M,G)$ does these give?
differential-geometry characteristic-classes de-rham-cohomology
asked Dec 27 '18 at 5:05
Praphulla Koushik
19515
This question has an open bounty worth +50
reputation from Praphulla Koushik ending in 3 days.
Looking for an answer drawing from credible and/or official sources.
add a comment |
This question is related to this question. The user who asked this question is not active since September. So, asking a separate question here.
Let $G$ be a compact Lie group and $Prightarrow M$ be a principal $G$ bundle. We want to associate Chern classes for this bundle.
This article says in its second page that "A compact Lie group $G$ may be shown (with some work, using the theory of compact operators) to be none other than a closed subgroup of the Unitary group $U(n)$ of matrices
for some $n$". Thus, for structure group $G$ of $P(M,G)$ there exists an embedding $Ghookrightarrow U(n)$.
See the embedding $Ghookrightarrow U(n)$ as an action of $G$ on $U(n)$. Given a manifold $P'$ and an action of $G$ on $P'$ there is a notion of what is called an associated bundle whose fibres are $P'$. In case $P'$ is a Lie group $H$, we get a principal $H$ bundle. This $G$ action on $U(n)$ gives a principal $U(n)$ bundle $Qrightarrow M$.
For this principal $U(n)$ bundle $Qrightarrow M$, we fix a connection $Gamma$ with curvature $Omega$. We then get what is called Weil homomorphism $W:I(U(n))rightarrow H^*(M,mathbb{R})$.
We choose elements $f_k:mathfrak{u}(n,mathbb{C})rightarrow mathbb{R}$ from $I(U(n))$ such that $$text{det}left(lambda I-frac{1}{2pisqrt{-1}}Xright)=sum_{k=0}^n f_k(X)lambda^{n-k}$$
The images $W(f_k)in H^{2k}(M,mathbb{R})$ are called the $k$-th Chern classes of $Qrightarrow M$.
How do we define Chern classes of the principal $G$ bundle $Prightarrow M$ that we have started with?
Is the $k$-th Chern class of $G$ bundle $P(M,G)$ the $k$-th Chern class of the $U(n)$ bundle $Q(M,U(n))$?
Does this depend on the embedding $Ghookrightarrow U(n)$ we have chosen?
In case it is dependent on the choice of $Ghookrightarrow U(n)$, what invariants of $P(M,G)$ does these give?
differential-geometry characteristic-classes de-rham-cohomology
asked Dec 27 '18 at 5:05
Praphulla Koushik
19515
This question has an open bounty worth +50
reputation from Praphulla Koushik ending in 3 days.
Looking for an answer drawing from credible and/or official sources.
add a comment |
This question is related to this question. The user who asked this question is not active since September. So, asking a separate question here.
Let $G$ be a compact Lie group and $Prightarrow M$ be a principal $G$ bundle. We want to associate Chern classes for this bundle.
This article says in its second page that "A compact Lie group $G$ may be shown (with some work, using the theory of compact operators) to be none other than a closed subgroup of the Unitary group $U(n)$ of matrices
for some $n$". Thus, for structure group $G$ of $P(M,G)$ there exists an embedding $Ghookrightarrow U(n)$.
See the embedding $Ghookrightarrow U(n)$ as an action of $G$ on $U(n)$. Given a manifold $P'$ and an action of $G$ on $P'$ there is a notion of what is called an associated bundle whose fibres are $P'$. In case $P'$ is a Lie group $H$, we get a principal $H$ bundle. This $G$ action on $U(n)$ gives a principal $U(n)$ bundle $Qrightarrow M$.
For this principal $U(n)$ bundle $Qrightarrow M$, we fix a connection $Gamma$ with curvature $Omega$. We then get what is called Weil homomorphism $W:I(U(n))rightarrow H^*(M,mathbb{R})$.
We choose elements $f_k:mathfrak{u}(n,mathbb{C})rightarrow mathbb{R}$ from $I(U(n))$ such that $$text{det}left(lambda I-frac{1}{2pisqrt{-1}}Xright)=sum_{k=0}^n f_k(X)lambda^{n-k}$$
The images $W(f_k)in H^{2k}(M,mathbb{R})$ are called the $k$-th Chern classes of $Qrightarrow M$.
How do we define Chern classes of the principal $G$ bundle $Prightarrow M$ that we have started with?
Is the $k$-th Chern class of $G$ bundle $P(M,G)$ the $k$-th Chern class of the $U(n)$ bundle $Q(M,U(n))$?
Does this depend on the embedding $Ghookrightarrow U(n)$ we have chosen?
In case it is dependent on the choice of $Ghookrightarrow U(n)$, what invariants of $P(M,G)$ does these give?
differential-geometry characteristic-classes de-rham-cohomology
asked Dec 27 '18 at 5:05
Praphulla Koushik
19515
This question is related to this question. The user who asked this question is not active since September. So, asking a separate question here.
Let $G$ be a compact Lie group and $Prightarrow M$ be a principal $G$ bundle. We want to associate Chern classes for this bundle.
This article says in its second page that "A compact Lie group $G$ may be shown (with some work, using the theory of compact operators) to be none other than a closed subgroup of the Unitary group $U(n)$ of matrices
for some $n$". Thus, for structure group $G$ of $P(M,G)$ there exists an embedding $Ghookrightarrow U(n)$.
See the embedding $Ghookrightarrow U(n)$ as an action of $G$ on $U(n)$. Given a manifold $P'$ and an action of $G$ on $P'$ there is a notion of what is called an associated bundle whose fibres are $P'$. In case $P'$ is a Lie group $H$, we get a principal $H$ bundle. This $G$ action on $U(n)$ gives a principal $U(n)$ bundle $Qrightarrow M$.
For this principal $U(n)$ bundle $Qrightarrow M$, we fix a connection $Gamma$ with curvature $Omega$. We then get what is called Weil homomorphism $W:I(U(n))rightarrow H^*(M,mathbb{R})$.
We choose elements $f_k:mathfrak{u}(n,mathbb{C})rightarrow mathbb{R}$ from $I(U(n))$ such that $$text{det}left(lambda I-frac{1}{2pisqrt{-1}}Xright)=sum_{k=0}^n f_k(X)lambda^{n-k}$$
The images $W(f_k)in H^{2k}(M,mathbb{R})$ are called the $k$-th Chern classes of $Qrightarrow M$.
How do we define Chern classes of the principal $G$ bundle $Prightarrow M$ that we have started with?
Is the $k$-th Chern class of $G$ bundle $P(M,G)$ the $k$-th Chern class of the $U(n)$ bundle $Q(M,U(n))$?
Does this depend on the embedding $Ghookrightarrow U(n)$ we have chosen?
In case it is dependent on the choice of $Ghookrightarrow U(n)$, what invariants of $P(M,G)$ does these give?
differential-geometry characteristic-classes de-rham-cohomology
differential-geometry characteristic-classes de-rham-cohomology
asked Dec 27 '18 at 5:05
Praphulla Koushik
19515
asked Dec 27 '18 at 5:05
Praphulla Koushik
19515
asked Dec 27 '18 at 5:05
Praphulla Koushik
19515
asked Dec 27 '18 at 5:05
Praphulla Koushik
19515
asked Dec 27 '18 at 5:05
Praphulla Koushik
19515
19515
This question has an open bounty worth +50
reputation from Praphulla Koushik ending in 3 days.
Looking for an answer drawing from credible and/or official sources.
This question has an open bounty worth +50
reputation from Praphulla Koushik ending in 3 days.
Looking for an answer drawing from credible and/or official sources.
add a comment |
add a comment |
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