Using Averaging and Deck Transformations to Compute de Rham Cohomologies of the $n$-Torus
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I'm preparing for some comprehensive exams and this is a question from a past year.
The $n$-torus $T^n = mathbbR^n/mathbbZ^n$ is both a smooth n-manifold and an Abelian group, by virtue of structures inherited from the real vector space $mathbbR^n$. In particular, $T^n$ acts on itself by smooth maps corresponding to translations of $mathbbR^n$. By averaging, prove that each de Rham cohomology class on $T^n$ contains a unique differential form which is translation-invariant. Then use this observation to calculate the dimensions of the de Rham cohomology vector spaces $H^p(T^n)$ for all $p$.
I can compute the de Rham cohomology using Kunneth's formula and some induction (I think the dimensions are something like $n choose p$) but that isn't the point. Plus, I'm quite curious about this idea of averaging so that we get something invariant under group actions. I've seen a similar question on showing that a Lens space is orientable by looking at some averaged form and then showing it is invariant under some orientation preserving transformations, or something like that.
differential-geometry differential-topology covering-spaces
edited Aug 11 at 16:43
John Ma
37.5k93669
asked Aug 11 at 0:12
inkievoyd
621316
add a comment |Â
up vote
3
down vote
favorite
I'm preparing for some comprehensive exams and this is a question from a past year.
The $n$-torus $T^n = mathbbR^n/mathbbZ^n$ is both a smooth n-manifold and an Abelian group, by virtue of structures inherited from the real vector space $mathbbR^n$. In particular, $T^n$ acts on itself by smooth maps corresponding to translations of $mathbbR^n$. By averaging, prove that each de Rham cohomology class on $T^n$ contains a unique differential form which is translation-invariant. Then use this observation to calculate the dimensions of the de Rham cohomology vector spaces $H^p(T^n)$ for all $p$.
I can compute the de Rham cohomology using Kunneth's formula and some induction (I think the dimensions are something like $n choose p$) but that isn't the point. Plus, I'm quite curious about this idea of averaging so that we get something invariant under group actions. I've seen a similar question on showing that a Lens space is orientable by looking at some averaged form and then showing it is invariant under some orientation preserving transformations, or something like that.
differential-geometry differential-topology covering-spaces
edited Aug 11 at 16:43
John Ma
37.5k93669
asked Aug 11 at 0:12
inkievoyd
621316
Here's a link that addresses your question: qchu.wordpress.com/2013/10/12/the-cohomology-of-the-n-torus It gives several methods for computing the cohomology and method 3 is relevant for you.
– Oliver Jones
Aug 11 at 1:25
Thank you for the link. Unfortunately, I have almost zero idea of what is going on, such as this result about Lie groups and Haar measures.
– inkievoyd
Aug 11 at 1:46
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I'm preparing for some comprehensive exams and this is a question from a past year.
The $n$-torus $T^n = mathbbR^n/mathbbZ^n$ is both a smooth n-manifold and an Abelian group, by virtue of structures inherited from the real vector space $mathbbR^n$. In particular, $T^n$ acts on itself by smooth maps corresponding to translations of $mathbbR^n$. By averaging, prove that each de Rham cohomology class on $T^n$ contains a unique differential form which is translation-invariant. Then use this observation to calculate the dimensions of the de Rham cohomology vector spaces $H^p(T^n)$ for all $p$.
I can compute the de Rham cohomology using Kunneth's formula and some induction (I think the dimensions are something like $n choose p$) but that isn't the point. Plus, I'm quite curious about this idea of averaging so that we get something invariant under group actions. I've seen a similar question on showing that a Lens space is orientable by looking at some averaged form and then showing it is invariant under some orientation preserving transformations, or something like that.
differential-geometry differential-topology covering-spaces
edited Aug 11 at 16:43
John Ma
37.5k93669
asked Aug 11 at 0:12
inkievoyd
621316
I'm preparing for some comprehensive exams and this is a question from a past year.
The $n$-torus $T^n = mathbbR^n/mathbbZ^n$ is both a smooth n-manifold and an Abelian group, by virtue of structures inherited from the real vector space $mathbbR^n$. In particular, $T^n$ acts on itself by smooth maps corresponding to translations of $mathbbR^n$. By averaging, prove that each de Rham cohomology class on $T^n$ contains a unique differential form which is translation-invariant. Then use this observation to calculate the dimensions of the de Rham cohomology vector spaces $H^p(T^n)$ for all $p$.
I can compute the de Rham cohomology using Kunneth's formula and some induction (I think the dimensions are something like $n choose p$) but that isn't the point. Plus, I'm quite curious about this idea of averaging so that we get something invariant under group actions. I've seen a similar question on showing that a Lens space is orientable by looking at some averaged form and then showing it is invariant under some orientation preserving transformations, or something like that.
differential-geometry differential-topology covering-spaces
edited Aug 11 at 16:43
John Ma
37.5k93669
asked Aug 11 at 0:12
inkievoyd
621316
edited Aug 11 at 16:43
John Ma
37.5k93669
edited Aug 11 at 16:43
John Ma
37.5k93669
edited Aug 11 at 16:43
John Ma
37.5k93669
37.5k93669
asked Aug 11 at 0:12
inkievoyd
621316
asked Aug 11 at 0:12
inkievoyd
621316
asked Aug 11 at 0:12
inkievoyd
621316
621316
Here's a link that addresses your question: qchu.wordpress.com/2013/10/12/the-cohomology-of-the-n-torus It gives several methods for computing the cohomology and method 3 is relevant for you.
– Oliver Jones
Aug 11 at 1:25
Thank you for the link. Unfortunately, I have almost zero idea of what is going on, such as this result about Lie groups and Haar measures.
– inkievoyd
Aug 11 at 1:46
add a comment |Â
Here's a link that addresses your question: qchu.wordpress.com/2013/10/12/the-cohomology-of-the-n-torus It gives several methods for computing the cohomology and method 3 is relevant for you.
– Oliver Jones
Aug 11 at 1:25
Thank you for the link. Unfortunately, I have almost zero idea of what is going on, such as this result about Lie groups and Haar measures.
– inkievoyd
Aug 11 at 1:46
Here's a link that addresses your question: qchu.wordpress.com/2013/10/12/the-cohomology-of-the-n-torus It gives several methods for computing the cohomology and method 3 is relevant for you.
– Oliver Jones
Aug 11 at 1:25
Here's a link that addresses your question: qchu.wordpress.com/2013/10/12/the-cohomology-of-the-n-torus It gives several methods for computing the cohomology and method 3 is relevant for you.
– Oliver Jones
Aug 11 at 1:25
Thank you for the link. Unfortunately, I have almost zero idea of what is going on, such as this result about Lie groups and Haar measures.
– inkievoyd
Aug 11 at 1:46
Thank you for the link. Unfortunately, I have almost zero idea of what is going on, such as this result about Lie groups and Haar measures.
– inkievoyd
Aug 11 at 1:46
add a comment |Â
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Here's a link that addresses your question: qchu.wordpress.com/2013/10/12/the-cohomology-of-the-n-torus It gives several methods for computing the cohomology and method 3 is relevant for you.
– Oliver Jones
Aug 11 at 1:25
Thank you for the link. Unfortunately, I have almost zero idea of what is going on, such as this result about Lie groups and Haar measures.
– inkievoyd
Aug 11 at 1:46