Two forms related by an automorphism are in the same cohomology class?












3












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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







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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
    6 hours ago
















3












$begingroup$


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







share|cite|improve this question









$endgroup$












  • $begingroup$
    Here's another type of example: Consider the antipodal map ($f(x)=-x$) on $S^n$ with $n$ even.
    $endgroup$
    – Ted Shifrin
    6 hours ago














3












3








3





$begingroup$


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







share|cite|improve this question









$endgroup$




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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




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asked 16 hours ago









jojojojo

5717




5717












  • $begingroup$
    Here's another type of example: Consider the antipodal map ($f(x)=-x$) on $S^n$ with $n$ even.
    $endgroup$
    – Ted Shifrin
    6 hours ago


















  • $begingroup$
    Here's another type of example: Consider the antipodal map ($f(x)=-x$) on $S^n$ with $n$ even.
    $endgroup$
    – Ted Shifrin
    6 hours ago
















$begingroup$
Here's another type of example: Consider the antipodal map ($f(x)=-x$) on $S^n$ with $n$ even.
$endgroup$
– Ted Shifrin
6 hours ago




$begingroup$
Here's another type of example: Consider the antipodal map ($f(x)=-x$) on $S^n$ with $n$ even.
$endgroup$
– Ted Shifrin
6 hours ago










1 Answer
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$begingroup$

No. One example: take the torus $X = mathbb{R}^2/mathbb{Z}^2$. The flip-flop on the factors interchanges the closed forms $dx$ and $dy$ which are linearly independent in $H^1(X)$.






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    1 Answer
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    7












    $begingroup$

    No. One example: take the torus $X = mathbb{R}^2/mathbb{Z}^2$. The flip-flop on the factors interchanges the closed forms $dx$ and $dy$ which are linearly independent in $H^1(X)$.






    share|cite|improve this answer









    $endgroup$


















      7












      $begingroup$

      No. One example: take the torus $X = mathbb{R}^2/mathbb{Z}^2$. The flip-flop on the factors interchanges the closed forms $dx$ and $dy$ which are linearly independent in $H^1(X)$.






      share|cite|improve this answer









      $endgroup$
















        7












        7








        7





        $begingroup$

        No. One example: take the torus $X = mathbb{R}^2/mathbb{Z}^2$. The flip-flop on the factors interchanges the closed forms $dx$ and $dy$ which are linearly independent in $H^1(X)$.






        share|cite|improve this answer









        $endgroup$



        No. One example: take the torus $X = mathbb{R}^2/mathbb{Z}^2$. The flip-flop on the factors interchanges the closed forms $dx$ and $dy$ which are linearly independent in $H^1(X)$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 16 hours ago









        hunterhunter

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