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