Two forms related by an automorphism are in the same cohomology class?
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Let $f: M to M$ define an automorphism on the smooth manifold M.
Given a differential form $omega in Omega^k$ is it true that the de Rham cohomology class of $omega$ and $f^*omega$ are the same? That is, does $[omega]=[f^*omega]$.
differential-geometry de-rham-cohomology
asked Mar 14 at 6:51
jojojojo
6217
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add a comment |
$begingroup$
Let $f: M to M$ define an automorphism on the smooth manifold M.
Given a differential form $omega in Omega^k$ is it true that the de Rham cohomology class of $omega$ and $f^*omega$ are the same? That is, does $[omega]=[f^*omega]$.
differential-geometry de-rham-cohomology
asked Mar 14 at 6:51
jojojojo
6217
$endgroup$
1
$begingroup$
Here's another type of example: Consider the antipodal map ($f(x)=-x$) on $S^n$ with $n$ even.
$endgroup$
– Ted Shifrin
Mar 14 at 16:29
add a comment |
$begingroup$
Let $f: M to M$ define an automorphism on the smooth manifold M.
Given a differential form $omega in Omega^k$ is it true that the de Rham cohomology class of $omega$ and $f^*omega$ are the same? That is, does $[omega]=[f^*omega]$.
differential-geometry de-rham-cohomology
asked Mar 14 at 6:51
jojojojo
6217
$endgroup$
Let $f: M to M$ define an automorphism on the smooth manifold M.
Given a differential form $omega in Omega^k$ is it true that the de Rham cohomology class of $omega$ and $f^*omega$ are the same? That is, does $[omega]=[f^*omega]$.
differential-geometry de-rham-cohomology
differential-geometry de-rham-cohomology
asked Mar 14 at 6:51
jojojojo
6217
asked Mar 14 at 6:51
jojojojo
6217
asked Mar 14 at 6:51
jojojojo
6217
asked Mar 14 at 6:51
jojojojo
6217
asked Mar 14 at 6:51
jojojojo
6217
6217
1
$begingroup$
Here's another type of example: Consider the antipodal map ($f(x)=-x$) on $S^n$ with $n$ even.
$endgroup$
– Ted Shifrin
Mar 14 at 16:29
add a comment |
1
$begingroup$
Here's another type of example: Consider the antipodal map ($f(x)=-x$) on $S^n$ with $n$ even.
$endgroup$
– Ted Shifrin
Mar 14 at 16:29
1
1
$begingroup$
Here's another type of example: Consider the antipodal map ($f(x)=-x$) on $S^n$ with $n$ even.
$endgroup$
– Ted Shifrin
Mar 14 at 16:29
$begingroup$
Here's another type of example: Consider the antipodal map ($f(x)=-x$) on $S^n$ with $n$ even.
$endgroup$
– Ted Shifrin
Mar 14 at 16:29
add a comment |
1 Answer
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No. One example: take the torus $X = mathbbR^2/mathbbZ^2$. The flip-flop on the factors interchanges the closed forms $dx$ and $dy$ which are linearly independent in $H^1(X)$.
answered Mar 14 at 6:54
hunterhunter
15.8k32643
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No. One example: take the torus $X = mathbbR^2/mathbbZ^2$. The flip-flop on the factors interchanges the closed forms $dx$ and $dy$ which are linearly independent in $H^1(X)$.
answered Mar 14 at 6:54
hunterhunter
15.8k32643
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add a comment |
$begingroup$
No. One example: take the torus $X = mathbbR^2/mathbbZ^2$. The flip-flop on the factors interchanges the closed forms $dx$ and $dy$ which are linearly independent in $H^1(X)$.
answered Mar 14 at 6:54
hunterhunter
15.8k32643
$endgroup$
add a comment |
$begingroup$
No. One example: take the torus $X = mathbbR^2/mathbbZ^2$. The flip-flop on the factors interchanges the closed forms $dx$ and $dy$ which are linearly independent in $H^1(X)$.
answered Mar 14 at 6:54
hunterhunter
15.8k32643
$endgroup$
No. One example: take the torus $X = mathbbR^2/mathbbZ^2$. The flip-flop on the factors interchanges the closed forms $dx$ and $dy$ which are linearly independent in $H^1(X)$.
answered Mar 14 at 6:54
hunterhunter
15.8k32643
answered Mar 14 at 6:54
hunterhunter
15.8k32643
answered Mar 14 at 6:54
hunterhunter
15.8k32643
answered Mar 14 at 6:54
hunterhunter
15.8k32643
15.8k32643
add a comment |
add a comment |
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$begingroup$
Here's another type of example: Consider the antipodal map ($f(x)=-x$) on $S^n$ with $n$ even.
$endgroup$
– Ted Shifrin
Mar 14 at 16:29