Recall that a manifold is called prime if whenever it is homeomorphic to a connected sum, one of the two summands is homeomorphic to a sphere.
It follows from the classification of surfaces that the closed prime surfaces are $S^2$, $T^2$, and $\mathbb{RP}^2$. Moreover, every closed surface decomposes as a connected sum of prime manifolds, and in the orientable case, the decomposition is unique up to reordering and $S^2$ summands (in the non-orientable case, one can restore uniqueness by prohibiting the use of $T^2$).
There is a similar story for closed three-manifolds: they always decompose as a connected sum of prime manifolds, and the decomposition is unique up to reordering and $S^3$ summands if the manifold is orientable, and uniqueness can be restored in the non-orientable case by prohibiting the use of $S^2\times S^1$. However, there are infinitely many prime three-manifolds. In the orientable case, they fit into three categories:
- those manifolds covered by $S^3$,
- the manifold $S^2\times S^1$, and
- orientable aspherical manifolds.
These categories can also be characterised via the fundamental group: namely finite, infinite cyclic, and infinite non-cyclic respectively.
Is there a similar categorisation of closed non-orientable prime three-manifolds?
One might suspect that such a categorisation immediately follow from the orientable case by passing to the orientable double cover. However, as Row shows in this paper, there exist closed non-orientable prime three-manifolds whose orientable double cover is not prime.
Best Answer
No, there is no such classification, there are way too many such manifolds. A standard way to construct these is to take a closed aspherical 3-manifold $L$ which admits an orientation-reversing involution with nonempty finite fixed-point set $\tau$. The quotient $L/\tau$ is an oribifold with even number of cone-points. Such orbifolds can be utterly wild, you can realize any finitely-presented group as its topological fundamental group:
Panov, D.; Petrunin, A., Telescopic actions, Geom. Funct. Anal. 22, No. 6, 1814-1831 (2012). ZBL1271.57051.
Cut out conical neighborhoods of these cone-points, you get a compact 3-manifold $M$ with even number of boundary $RP^2$'s. Glue them in pairs to get a closed 3-manifold $N$. One can show that $N$ is prime but it is far from aspherical. More generally, you can take several manifolds $M$ like that and glue them along boundary projective planes. This construction is a little dirty secret of 3D topology, which is why people in the field prefer to work with oriented manifolds. It also explains that in order to truly geometrize nonorientable 3-manifolds, one is forced to enlarge the category and work with orbifolds: In this category one modifies the notion of the connected sum by allowing removing not only balls but orbi-balls as well. Accordingly, the notion of a primeness has to be modified as well. This is strangely similar to the MMP in algebraic geometry.