Given a 3-manifold $M$, what algebraic-topological methods can one use to detect or obstruct whether or not it is an orientation double cover?
Some basic things one can quickly observe about orientation double covers $M$ include:
1) M admits a free, orientation reversing, involutive homeomorphism
2) By the Lefshetz fixed point theorem, if $\phi$ is the homeomorphism from 1), one has $$ 1 - tr(\phi_{*}|_{H_{1}}) + tr(\phi_{*}|_{H_{2}}) + 1 =0$$ This implies for example that rational homology 3-spheres don't arise as orientation double covers, and that orientation double covers have infinite $H_{1}$ and $H_{2}$. Perhaps it has other implications I don't see?
Are there other basic facts or results which allow you to look at algebraic-topological data associated to $M$ to obstruct it being such a cover?