A direct proof of the Chern-Weil isomorphism

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Given a principal $G$-bundle $P to M := P / G$ with Lie group $G$ and associated Lie algebra $g$, the Chern-Weil homomorphism
$$S^*(g)^G to H_DR^*(M)$$
associated to any invariant polynomial on $g$, a characteristic class in the de Rham cohomology of $M$. If $G$ is compact and the bundle is the universal bundle $E_G to B_G$, then the Chern-Weil homomorphism can be shown to be an isomorphism.

So far I have seen in the literature only the construction of the homomorphism for smooth finite dimensional principal bundles, and one construction and proof of the isomorphism in the case of classifying spaces of compact Lie groups in Dupont's book "Curvature and Characteristic Classes". In this book, the author introduces the notion of simplicial manifolds and their de Rham cohomology, in order to work through the fact that $B_G$ is not a smooth finite dimensional manifold, and therefore differential forms are not well-defined a priori.

My question is the following: are there any more direct constructions of the Chern-Weil isomorphism in the case where $M = B_G$ ?

I am thinking of the following: we could filtrate $E_G to B_G$ by smooth finite dimensional principal bundle $E_jG to B_j G$ on which $G$ acts freely, so that we get a homomorphism at each stage $j$, and then conclude at the limit, defining a differential form on $B_G$ as a class in the limit $undersetjlim Omega^*(B_j G)$. It seems to work for the universal bundle $S^infty to mathbbC^infty$ for $G = S^1$, but I might be missing something.

Thanks a lot

characteristic-classes classifying-spaces de-rham-cohomology

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asked Jul 27 at 11:20

BrianT

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Given a principal $G$-bundle $P to M := P / G$ with Lie group $G$ and associated Lie algebra $g$, the Chern-Weil homomorphism
$$S^*(g)^G to H_DR^*(M)$$
associated to any invariant polynomial on $g$, a characteristic class in the de Rham cohomology of $M$. If $G$ is compact and the bundle is the universal bundle $E_G to B_G$, then the Chern-Weil homomorphism can be shown to be an isomorphism.

So far I have seen in the literature only the construction of the homomorphism for smooth finite dimensional principal bundles, and one construction and proof of the isomorphism in the case of classifying spaces of compact Lie groups in Dupont's book "Curvature and Characteristic Classes". In this book, the author introduces the notion of simplicial manifolds and their de Rham cohomology, in order to work through the fact that $B_G$ is not a smooth finite dimensional manifold, and therefore differential forms are not well-defined a priori.

My question is the following: are there any more direct constructions of the Chern-Weil isomorphism in the case where $M = B_G$ ?

I am thinking of the following: we could filtrate $E_G to B_G$ by smooth finite dimensional principal bundle $E_jG to B_j G$ on which $G$ acts freely, so that we get a homomorphism at each stage $j$, and then conclude at the limit, defining a differential form on $B_G$ as a class in the limit $undersetjlim Omega^*(B_j G)$. It seems to work for the universal bundle $S^infty to mathbbC^infty$ for $G = S^1$, but I might be missing something.

Thanks a lot

characteristic-classes classifying-spaces de-rham-cohomology

share|cite|improve this question

asked Jul 27 at 11:20

BrianT

214

add a comment | 

up vote
0
down vote

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up vote
0
down vote

favorite

Given a principal $G$-bundle $P to M := P / G$ with Lie group $G$ and associated Lie algebra $g$, the Chern-Weil homomorphism
$$S^*(g)^G to H_DR^*(M)$$
associated to any invariant polynomial on $g$, a characteristic class in the de Rham cohomology of $M$. If $G$ is compact and the bundle is the universal bundle $E_G to B_G$, then the Chern-Weil homomorphism can be shown to be an isomorphism.

So far I have seen in the literature only the construction of the homomorphism for smooth finite dimensional principal bundles, and one construction and proof of the isomorphism in the case of classifying spaces of compact Lie groups in Dupont's book "Curvature and Characteristic Classes". In this book, the author introduces the notion of simplicial manifolds and their de Rham cohomology, in order to work through the fact that $B_G$ is not a smooth finite dimensional manifold, and therefore differential forms are not well-defined a priori.

My question is the following: are there any more direct constructions of the Chern-Weil isomorphism in the case where $M = B_G$ ?

I am thinking of the following: we could filtrate $E_G to B_G$ by smooth finite dimensional principal bundle $E_jG to B_j G$ on which $G$ acts freely, so that we get a homomorphism at each stage $j$, and then conclude at the limit, defining a differential form on $B_G$ as a class in the limit $undersetjlim Omega^*(B_j G)$. It seems to work for the universal bundle $S^infty to mathbbC^infty$ for $G = S^1$, but I might be missing something.

Thanks a lot

characteristic-classes classifying-spaces de-rham-cohomology

share|cite|improve this question

asked Jul 27 at 11:20

BrianT

214

Given a principal $G$-bundle $P to M := P / G$ with Lie group $G$ and associated Lie algebra $g$, the Chern-Weil homomorphism
$$S^*(g)^G to H_DR^*(M)$$
associated to any invariant polynomial on $g$, a characteristic class in the de Rham cohomology of $M$. If $G$ is compact and the bundle is the universal bundle $E_G to B_G$, then the Chern-Weil homomorphism can be shown to be an isomorphism.

So far I have seen in the literature only the construction of the homomorphism for smooth finite dimensional principal bundles, and one construction and proof of the isomorphism in the case of classifying spaces of compact Lie groups in Dupont's book "Curvature and Characteristic Classes". In this book, the author introduces the notion of simplicial manifolds and their de Rham cohomology, in order to work through the fact that $B_G$ is not a smooth finite dimensional manifold, and therefore differential forms are not well-defined a priori.

My question is the following: are there any more direct constructions of the Chern-Weil isomorphism in the case where $M = B_G$ ?

I am thinking of the following: we could filtrate $E_G to B_G$ by smooth finite dimensional principal bundle $E_jG to B_j G$ on which $G$ acts freely, so that we get a homomorphism at each stage $j$, and then conclude at the limit, defining a differential form on $B_G$ as a class in the limit $undersetjlim Omega^*(B_j G)$. It seems to work for the universal bundle $S^infty to mathbbC^infty$ for $G = S^1$, but I might be missing something.

Thanks a lot

characteristic-classes classifying-spaces de-rham-cohomology

share|cite|improve this question

asked Jul 27 at 11:20

BrianT

214

share|cite|improve this question

share|cite|improve this question

asked Jul 27 at 11:20

BrianT

214

asked Jul 27 at 11:20

BrianT

214

asked Jul 27 at 11:20

BrianT

214

214

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