$F:Mto N$ is surjective if $int_M F^* eta ne 0$ for some $eta in Omega^n(N)$












4












$begingroup$


Let $M$ and $N$ be compact orientable and connected smooth $n$-manifolds and $F:M to N$ a smooth map. Suppose $$int_M F^* eta ne 0$$ for some $eta in Omega^n(N)$. Then $F$ is surjective. Give an example that shows the converse is not true.




A non-surjective map has degree $0$ so the first part is clear. I could not think of an example for the converse, however. I want to find two compact oriented connected manifolds such that $F$ is surjective but $int_M F^* eta = 0$ for all $eta in Omega^n(N)$.






differential-geometry smooth-manifolds de-rham-cohomology







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    @MoisheCohen Yes I'm looking for a counterexample
    $endgroup$
    – mysatellite
    Dec 31 '18 at 2:01






  • 1




    $begingroup$
    Choose $F$ surjective but null-homotopic.
    $endgroup$
    – user98602
    Dec 31 '18 at 2:20










  • $begingroup$
    I don't know if you read the deleted answer and all the comments therein before it was deleted. Are you still interested in an answer?
    $endgroup$
    – Amitai Yuval
    Dec 31 '18 at 16:24










  • $begingroup$
    @AmitaiYuval yes
    $endgroup$
    – mysatellite
    Dec 31 '18 at 17:11
















4












$begingroup$


Let $M$ and $N$ be compact orientable and connected smooth $n$-manifolds and $F:M to N$ a smooth map. Suppose $$int_M F^* eta ne 0$$ for some $eta in Omega^n(N)$. Then $F$ is surjective. Give an example that shows the converse is not true.




A non-surjective map has degree $0$ so the first part is clear. I could not think of an example for the converse, however. I want to find two compact oriented connected manifolds such that $F$ is surjective but $int_M F^* eta = 0$ for all $eta in Omega^n(N)$.






differential-geometry smooth-manifolds de-rham-cohomology







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    @MoisheCohen Yes I'm looking for a counterexample
    $endgroup$
    – mysatellite
    Dec 31 '18 at 2:01






  • 1




    $begingroup$
    Choose $F$ surjective but null-homotopic.
    $endgroup$
    – user98602
    Dec 31 '18 at 2:20










  • $begingroup$
    I don't know if you read the deleted answer and all the comments therein before it was deleted. Are you still interested in an answer?
    $endgroup$
    – Amitai Yuval
    Dec 31 '18 at 16:24










  • $begingroup$
    @AmitaiYuval yes
    $endgroup$
    – mysatellite
    Dec 31 '18 at 17:11














4












4








4





$begingroup$


Let $M$ and $N$ be compact orientable and connected smooth $n$-manifolds and $F:M to N$ a smooth map. Suppose $$int_M F^* eta ne 0$$ for some $eta in Omega^n(N)$. Then $F$ is surjective. Give an example that shows the converse is not true.




A non-surjective map has degree $0$ so the first part is clear. I could not think of an example for the converse, however. I want to find two compact oriented connected manifolds such that $F$ is surjective but $int_M F^* eta = 0$ for all $eta in Omega^n(N)$.






differential-geometry smooth-manifolds de-rham-cohomology







share|cite|improve this question









$endgroup$




Let $M$ and $N$ be compact orientable and connected smooth $n$-manifolds and $F:M to N$ a smooth map. Suppose $$int_M F^* eta ne 0$$ for some $eta in Omega^n(N)$. Then $F$ is surjective. Give an example that shows the converse is not true.




A non-surjective map has degree $0$ so the first part is clear. I could not think of an example for the converse, however. I want to find two compact oriented connected manifolds such that $F$ is surjective but $int_M F^* eta = 0$ for all $eta in Omega^n(N)$.






differential-geometry smooth-manifolds de-rham-cohomology




differential-geometry smooth-manifolds de-rham-cohomology






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 31 '18 at 1:44









mysatellitemysatellite

2,14221231




2,14221231








  • 1




    $begingroup$
    @MoisheCohen Yes I'm looking for a counterexample
    $endgroup$
    – mysatellite
    Dec 31 '18 at 2:01






  • 1




    $begingroup$
    Choose $F$ surjective but null-homotopic.
    $endgroup$
    – user98602
    Dec 31 '18 at 2:20










  • $begingroup$
    I don't know if you read the deleted answer and all the comments therein before it was deleted. Are you still interested in an answer?
    $endgroup$
    – Amitai Yuval
    Dec 31 '18 at 16:24










  • $begingroup$
    @AmitaiYuval yes
    $endgroup$
    – mysatellite
    Dec 31 '18 at 17:11














  • 1




    $begingroup$
    @MoisheCohen Yes I'm looking for a counterexample
    $endgroup$
    – mysatellite
    Dec 31 '18 at 2:01






  • 1




    $begingroup$
    Choose $F$ surjective but null-homotopic.
    $endgroup$
    – user98602
    Dec 31 '18 at 2:20










  • $begingroup$
    I don't know if you read the deleted answer and all the comments therein before it was deleted. Are you still interested in an answer?
    $endgroup$
    – Amitai Yuval
    Dec 31 '18 at 16:24










  • $begingroup$
    @AmitaiYuval yes
    $endgroup$
    – mysatellite
    Dec 31 '18 at 17:11








1




1




$begingroup$
@MoisheCohen Yes I'm looking for a counterexample
$endgroup$
– mysatellite
Dec 31 '18 at 2:01




$begingroup$
@MoisheCohen Yes I'm looking for a counterexample
$endgroup$
– mysatellite
Dec 31 '18 at 2:01




1




1




$begingroup$
Choose $F$ surjective but null-homotopic.
$endgroup$
– user98602
Dec 31 '18 at 2:20




$begingroup$
Choose $F$ surjective but null-homotopic.
$endgroup$
– user98602
Dec 31 '18 at 2:20












$begingroup$
I don't know if you read the deleted answer and all the comments therein before it was deleted. Are you still interested in an answer?
$endgroup$
– Amitai Yuval
Dec 31 '18 at 16:24




$begingroup$
I don't know if you read the deleted answer and all the comments therein before it was deleted. Are you still interested in an answer?
$endgroup$
– Amitai Yuval
Dec 31 '18 at 16:24












$begingroup$
@AmitaiYuval yes
$endgroup$
– mysatellite
Dec 31 '18 at 17:11




$begingroup$
@AmitaiYuval yes
$endgroup$
– mysatellite
Dec 31 '18 at 17:11










1 Answer
1






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oldest

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2












$begingroup$

Here is a concrete realization of Mike Miller's comment. Think of $S^1$ as sitting in $mathbb{C}$ and consider the map
begin{align*}
varphi: S^1 & to S^1 \
x+iy & mapsto e^{2pi i x}.
end{align*}
Then $varphi$ is both surjective and null-homotopic and thus serves as a counterexample.






share|cite|improve this answer









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    1 Answer
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    1 Answer
    1






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    active

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    active

    oldest

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    2












    $begingroup$

    Here is a concrete realization of Mike Miller's comment. Think of $S^1$ as sitting in $mathbb{C}$ and consider the map
    begin{align*}
    varphi: S^1 & to S^1 \
    x+iy & mapsto e^{2pi i x}.
    end{align*}
    Then $varphi$ is both surjective and null-homotopic and thus serves as a counterexample.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      Here is a concrete realization of Mike Miller's comment. Think of $S^1$ as sitting in $mathbb{C}$ and consider the map
      begin{align*}
      varphi: S^1 & to S^1 \
      x+iy & mapsto e^{2pi i x}.
      end{align*}
      Then $varphi$ is both surjective and null-homotopic and thus serves as a counterexample.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        Here is a concrete realization of Mike Miller's comment. Think of $S^1$ as sitting in $mathbb{C}$ and consider the map
        begin{align*}
        varphi: S^1 & to S^1 \
        x+iy & mapsto e^{2pi i x}.
        end{align*}
        Then $varphi$ is both surjective and null-homotopic and thus serves as a counterexample.






        share|cite|improve this answer









        $endgroup$



        Here is a concrete realization of Mike Miller's comment. Think of $S^1$ as sitting in $mathbb{C}$ and consider the map
        begin{align*}
        varphi: S^1 & to S^1 \
        x+iy & mapsto e^{2pi i x}.
        end{align*}
        Then $varphi$ is both surjective and null-homotopic and thus serves as a counterexample.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 31 '18 at 21:08









        Or EisenbergOr Eisenberg

        1596




        1596






























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