Manifolds with nonwhere vanishing closed one formsVanishing of Euler classDoes a connected manifold with vanishing Euler characteristic admit a nowhere-vanishing vector field?Homotopy spheres with vanishing and non-vanishing $alpha$-invariantNowhere vanishing, normalized vector field with bounded derivativesVanishing of characteristic numbers vs vanishing of characteristic classesNon-orientable closed 4- or 3-manifolds with known cohomology ringExamples of interesting non orientable closed 3-manifolds
Manifolds with nonwhere vanishing closed one forms
Vanishing of Euler classDoes a connected manifold with vanishing Euler characteristic admit a nowhere-vanishing vector field?Homotopy spheres with vanishing and non-vanishing $alpha$-invariantNowhere vanishing, normalized vector field with bounded derivativesVanishing of characteristic numbers vs vanishing of characteristic classesNon-orientable closed 4- or 3-manifolds with known cohomology ringExamples of interesting non orientable closed 3-manifolds
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I am trying to find examples of closed manifolds $M$ admitting a nowhere vanishing closed one form. I am wondering if there are any examples beyond $Ntimes S^1$.
at.algebraic-topology smooth-manifolds examples
asked Oct 2 at 0:59
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I am trying to find examples of closed manifolds $M$ admitting a nowhere vanishing closed one form. I am wondering if there are any examples beyond $Ntimes S^1$.
at.algebraic-topology smooth-manifolds examples
asked Oct 2 at 0:59
ZZYZZY
5632 silver badges9 bronze badges
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add a comment
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$begingroup$
I am trying to find examples of closed manifolds $M$ admitting a nowhere vanishing closed one form. I am wondering if there are any examples beyond $Ntimes S^1$.
at.algebraic-topology smooth-manifolds examples
asked Oct 2 at 0:59
ZZYZZY
5632 silver badges9 bronze badges
$endgroup$
I am trying to find examples of closed manifolds $M$ admitting a nowhere vanishing closed one form. I am wondering if there are any examples beyond $Ntimes S^1$.
at.algebraic-topology smooth-manifolds examples
at.algebraic-topology smooth-manifolds examples
asked Oct 2 at 0:59
ZZYZZY
5632 silver badges9 bronze badges
asked Oct 2 at 0:59
ZZYZZY
5632 silver badges9 bronze badges
asked Oct 2 at 0:59
ZZYZZY
5632 silver badges9 bronze badges
asked Oct 2 at 0:59
ZZYZZY
5632 silver badges9 bronze badges
asked Oct 2 at 0:59
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1 Answer
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If $f: M to S^1$ is a submersion (and so a fiber bundle if $M$ is compact) then $f^*dtheta$ is a nowhere-vanishing closed 1-form. There are many more such manifolds and fibrations than just products, and the manifolds have the name mapping tori.
If $(M,omega)$ is a manifold equipped with a nowhere vanishing closed 1-form, then in fact there is a fibration over the circle $f: M to S^1$ so that $f^*dtheta$ is arbitrarily close to $omega$. This is a theorem of Tischler, whose short proof can be read here.
answered Oct 2 at 1:43
Mike MillerMike Miller
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1 Answer
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1 Answer
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$begingroup$
If $f: M to S^1$ is a submersion (and so a fiber bundle if $M$ is compact) then $f^*dtheta$ is a nowhere-vanishing closed 1-form. There are many more such manifolds and fibrations than just products, and the manifolds have the name mapping tori.
If $(M,omega)$ is a manifold equipped with a nowhere vanishing closed 1-form, then in fact there is a fibration over the circle $f: M to S^1$ so that $f^*dtheta$ is arbitrarily close to $omega$. This is a theorem of Tischler, whose short proof can be read here.
answered Oct 2 at 1:43
Mike MillerMike Miller
5,5835 gold badges28 silver badges47 bronze badges
$endgroup$
add a comment
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$begingroup$
If $f: M to S^1$ is a submersion (and so a fiber bundle if $M$ is compact) then $f^*dtheta$ is a nowhere-vanishing closed 1-form. There are many more such manifolds and fibrations than just products, and the manifolds have the name mapping tori.
If $(M,omega)$ is a manifold equipped with a nowhere vanishing closed 1-form, then in fact there is a fibration over the circle $f: M to S^1$ so that $f^*dtheta$ is arbitrarily close to $omega$. This is a theorem of Tischler, whose short proof can be read here.
answered Oct 2 at 1:43
Mike MillerMike Miller
5,5835 gold badges28 silver badges47 bronze badges
$endgroup$
add a comment
|
$begingroup$
If $f: M to S^1$ is a submersion (and so a fiber bundle if $M$ is compact) then $f^*dtheta$ is a nowhere-vanishing closed 1-form. There are many more such manifolds and fibrations than just products, and the manifolds have the name mapping tori.
If $(M,omega)$ is a manifold equipped with a nowhere vanishing closed 1-form, then in fact there is a fibration over the circle $f: M to S^1$ so that $f^*dtheta$ is arbitrarily close to $omega$. This is a theorem of Tischler, whose short proof can be read here.
answered Oct 2 at 1:43
Mike MillerMike Miller
5,5835 gold badges28 silver badges47 bronze badges
$endgroup$
If $f: M to S^1$ is a submersion (and so a fiber bundle if $M$ is compact) then $f^*dtheta$ is a nowhere-vanishing closed 1-form. There are many more such manifolds and fibrations than just products, and the manifolds have the name mapping tori.
If $(M,omega)$ is a manifold equipped with a nowhere vanishing closed 1-form, then in fact there is a fibration over the circle $f: M to S^1$ so that $f^*dtheta$ is arbitrarily close to $omega$. This is a theorem of Tischler, whose short proof can be read here.
answered Oct 2 at 1:43
Mike MillerMike Miller
5,5835 gold badges28 silver badges47 bronze badges
answered Oct 2 at 1:43
Mike MillerMike Miller
5,5835 gold badges28 silver badges47 bronze badges
answered Oct 2 at 1:43
Mike MillerMike Miller
5,5835 gold badges28 silver badges47 bronze badges
answered Oct 2 at 1:43
Mike MillerMike Miller
5,5835 gold badges28 silver badges47 bronze badges
5,5835 gold badges28 silver badges47 bronze badges
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