Which Nonorientable 3 manifolds have torsion in $H_{1}$?












5












$begingroup$


In thinking about closed, nonorientable 3 manifolds I've been interested in finding nonorientable loops which represent torsion classes in homology. More precisely, for a closed nonorientable 3 manifold $M$ I've wondered when one can find a class $gamma in H_{1}(M; mathbb{Z})$ so that $gamma$ is torsion, and $w_{1}(M)[bar{gamma}]$ is nontrivial (where $w_{1}(M)$ is the first Stiefel-Whitney class of the tangent bundle and $bar{gamma}$ is the reduction mod 2 of $gamma$).



This has led me to the following questions:





Can one characterize which nonorientable 3 manifolds $M$ have torsion classes in $H_{1}(M ; mathbb{Z})$?





My list of examples of nonorientable 3 manifolds is short, but among them I've noticed that some do:



1) $S^{1} times N_{h}$, where $N_{h} cong (mathbb{R}P^{2})^{#h}$



and some don't:



2) $S^{1} tilde{times} S^{2}$



Secondly,





For those 3 manifolds $M$ with torsion in $H_{1}(M; mathbb{Z})$, can one characterize which have torsion classes $gamma$ satisfying $w_{1}(M)[bar gamma]$ =1?





I've noted here as well that some do:



$S^{1} times N_{1}$



and some don't:



$S^{1} times N_{2}$.






general-topology differential-geometry algebraic-topology differential-topology geometric-topology







share|cite|improve this question









$endgroup$












  • $begingroup$
    I don't know the answer but I'm guessing that if such $gamma$ exists in $M$, then it also exists in $M # M'$, which gives more examples.
    $endgroup$
    – Hew Wolff
    Dec 4 '18 at 3:29








  • 3




    $begingroup$
    I am skeptical there is any characterization.
    $endgroup$
    – Mike Miller
    Dec 4 '18 at 20:46










  • $begingroup$
    @HewWolff, I agree; I think this should follow from additivity of $w_{1}$ under connect sums, but haven't yet written this down.
    $endgroup$
    – It'sRecreational
    Dec 5 '18 at 0:21












  • $begingroup$
    @MikeMiller I'm similarly skeptical, but don't really have much evidence for skepticism or optimism. Do you have more classes of examples in mind or some justification for this intuition? My feeling is that the second condition is more likely 'characterizable'.
    $endgroup$
    – It'sRecreational
    Dec 5 '18 at 0:26
















5












$begingroup$


In thinking about closed, nonorientable 3 manifolds I've been interested in finding nonorientable loops which represent torsion classes in homology. More precisely, for a closed nonorientable 3 manifold $M$ I've wondered when one can find a class $gamma in H_{1}(M; mathbb{Z})$ so that $gamma$ is torsion, and $w_{1}(M)[bar{gamma}]$ is nontrivial (where $w_{1}(M)$ is the first Stiefel-Whitney class of the tangent bundle and $bar{gamma}$ is the reduction mod 2 of $gamma$).



This has led me to the following questions:





Can one characterize which nonorientable 3 manifolds $M$ have torsion classes in $H_{1}(M ; mathbb{Z})$?





My list of examples of nonorientable 3 manifolds is short, but among them I've noticed that some do:



1) $S^{1} times N_{h}$, where $N_{h} cong (mathbb{R}P^{2})^{#h}$



and some don't:



2) $S^{1} tilde{times} S^{2}$



Secondly,





For those 3 manifolds $M$ with torsion in $H_{1}(M; mathbb{Z})$, can one characterize which have torsion classes $gamma$ satisfying $w_{1}(M)[bar gamma]$ =1?





I've noted here as well that some do:



$S^{1} times N_{1}$



and some don't:



$S^{1} times N_{2}$.






general-topology differential-geometry algebraic-topology differential-topology geometric-topology







share|cite|improve this question









$endgroup$












  • $begingroup$
    I don't know the answer but I'm guessing that if such $gamma$ exists in $M$, then it also exists in $M # M'$, which gives more examples.
    $endgroup$
    – Hew Wolff
    Dec 4 '18 at 3:29








  • 3




    $begingroup$
    I am skeptical there is any characterization.
    $endgroup$
    – Mike Miller
    Dec 4 '18 at 20:46










  • $begingroup$
    @HewWolff, I agree; I think this should follow from additivity of $w_{1}$ under connect sums, but haven't yet written this down.
    $endgroup$
    – It'sRecreational
    Dec 5 '18 at 0:21












  • $begingroup$
    @MikeMiller I'm similarly skeptical, but don't really have much evidence for skepticism or optimism. Do you have more classes of examples in mind or some justification for this intuition? My feeling is that the second condition is more likely 'characterizable'.
    $endgroup$
    – It'sRecreational
    Dec 5 '18 at 0:26














5












5








5





$begingroup$


In thinking about closed, nonorientable 3 manifolds I've been interested in finding nonorientable loops which represent torsion classes in homology. More precisely, for a closed nonorientable 3 manifold $M$ I've wondered when one can find a class $gamma in H_{1}(M; mathbb{Z})$ so that $gamma$ is torsion, and $w_{1}(M)[bar{gamma}]$ is nontrivial (where $w_{1}(M)$ is the first Stiefel-Whitney class of the tangent bundle and $bar{gamma}$ is the reduction mod 2 of $gamma$).



This has led me to the following questions:





Can one characterize which nonorientable 3 manifolds $M$ have torsion classes in $H_{1}(M ; mathbb{Z})$?





My list of examples of nonorientable 3 manifolds is short, but among them I've noticed that some do:



1) $S^{1} times N_{h}$, where $N_{h} cong (mathbb{R}P^{2})^{#h}$



and some don't:



2) $S^{1} tilde{times} S^{2}$



Secondly,





For those 3 manifolds $M$ with torsion in $H_{1}(M; mathbb{Z})$, can one characterize which have torsion classes $gamma$ satisfying $w_{1}(M)[bar gamma]$ =1?





I've noted here as well that some do:



$S^{1} times N_{1}$



and some don't:



$S^{1} times N_{2}$.






general-topology differential-geometry algebraic-topology differential-topology geometric-topology







share|cite|improve this question









$endgroup$




In thinking about closed, nonorientable 3 manifolds I've been interested in finding nonorientable loops which represent torsion classes in homology. More precisely, for a closed nonorientable 3 manifold $M$ I've wondered when one can find a class $gamma in H_{1}(M; mathbb{Z})$ so that $gamma$ is torsion, and $w_{1}(M)[bar{gamma}]$ is nontrivial (where $w_{1}(M)$ is the first Stiefel-Whitney class of the tangent bundle and $bar{gamma}$ is the reduction mod 2 of $gamma$).



This has led me to the following questions:





Can one characterize which nonorientable 3 manifolds $M$ have torsion classes in $H_{1}(M ; mathbb{Z})$?





My list of examples of nonorientable 3 manifolds is short, but among them I've noticed that some do:



1) $S^{1} times N_{h}$, where $N_{h} cong (mathbb{R}P^{2})^{#h}$



and some don't:



2) $S^{1} tilde{times} S^{2}$



Secondly,





For those 3 manifolds $M$ with torsion in $H_{1}(M; mathbb{Z})$, can one characterize which have torsion classes $gamma$ satisfying $w_{1}(M)[bar gamma]$ =1?





I've noted here as well that some do:



$S^{1} times N_{1}$



and some don't:



$S^{1} times N_{2}$.






general-topology differential-geometry algebraic-topology differential-topology geometric-topology




general-topology differential-geometry algebraic-topology differential-topology geometric-topology






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 4 '18 at 1:17









It'sRecreationalIt'sRecreational

906




906












  • $begingroup$
    I don't know the answer but I'm guessing that if such $gamma$ exists in $M$, then it also exists in $M # M'$, which gives more examples.
    $endgroup$
    – Hew Wolff
    Dec 4 '18 at 3:29








  • 3




    $begingroup$
    I am skeptical there is any characterization.
    $endgroup$
    – Mike Miller
    Dec 4 '18 at 20:46










  • $begingroup$
    @HewWolff, I agree; I think this should follow from additivity of $w_{1}$ under connect sums, but haven't yet written this down.
    $endgroup$
    – It'sRecreational
    Dec 5 '18 at 0:21












  • $begingroup$
    @MikeMiller I'm similarly skeptical, but don't really have much evidence for skepticism or optimism. Do you have more classes of examples in mind or some justification for this intuition? My feeling is that the second condition is more likely 'characterizable'.
    $endgroup$
    – It'sRecreational
    Dec 5 '18 at 0:26


















  • $begingroup$
    I don't know the answer but I'm guessing that if such $gamma$ exists in $M$, then it also exists in $M # M'$, which gives more examples.
    $endgroup$
    – Hew Wolff
    Dec 4 '18 at 3:29








  • 3




    $begingroup$
    I am skeptical there is any characterization.
    $endgroup$
    – Mike Miller
    Dec 4 '18 at 20:46










  • $begingroup$
    @HewWolff, I agree; I think this should follow from additivity of $w_{1}$ under connect sums, but haven't yet written this down.
    $endgroup$
    – It'sRecreational
    Dec 5 '18 at 0:21












  • $begingroup$
    @MikeMiller I'm similarly skeptical, but don't really have much evidence for skepticism or optimism. Do you have more classes of examples in mind or some justification for this intuition? My feeling is that the second condition is more likely 'characterizable'.
    $endgroup$
    – It'sRecreational
    Dec 5 '18 at 0:26
















$begingroup$
I don't know the answer but I'm guessing that if such $gamma$ exists in $M$, then it also exists in $M # M'$, which gives more examples.
$endgroup$
– Hew Wolff
Dec 4 '18 at 3:29






$begingroup$
I don't know the answer but I'm guessing that if such $gamma$ exists in $M$, then it also exists in $M # M'$, which gives more examples.
$endgroup$
– Hew Wolff
Dec 4 '18 at 3:29






3




3




$begingroup$
I am skeptical there is any characterization.
$endgroup$
– Mike Miller
Dec 4 '18 at 20:46




$begingroup$
I am skeptical there is any characterization.
$endgroup$
– Mike Miller
Dec 4 '18 at 20:46












$begingroup$
@HewWolff, I agree; I think this should follow from additivity of $w_{1}$ under connect sums, but haven't yet written this down.
$endgroup$
– It'sRecreational
Dec 5 '18 at 0:21






$begingroup$
@HewWolff, I agree; I think this should follow from additivity of $w_{1}$ under connect sums, but haven't yet written this down.
$endgroup$
– It'sRecreational
Dec 5 '18 at 0:21














$begingroup$
@MikeMiller I'm similarly skeptical, but don't really have much evidence for skepticism or optimism. Do you have more classes of examples in mind or some justification for this intuition? My feeling is that the second condition is more likely 'characterizable'.
$endgroup$
– It'sRecreational
Dec 5 '18 at 0:26




$begingroup$
@MikeMiller I'm similarly skeptical, but don't really have much evidence for skepticism or optimism. Do you have more classes of examples in mind or some justification for this intuition? My feeling is that the second condition is more likely 'characterizable'.
$endgroup$
– It'sRecreational
Dec 5 '18 at 0:26










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