Non-orientable real algebraic three-dimensional manifolds

Question

Smooth real algebraic hypersurfaces of even degree in $\mathbb{RP}^4$ that are maximal (i.e. that are homologically as rich as possible in the sense of the Smith-Thom inequality) are all non-orientable (see Section 1.2 in $\textit{Mutual position of hypersurfaces in projective space}$, O. Viro ).

I am trying to investigate the possible topological types and geometries of such hypersurfaces, but all the sources about 3-dimensional manifolds I browsed reduce to the orientable case and say that analogue results can be proven in the non-orientable with some adjustments. Do you know where I could find precise statements on this topic ?

In particular, I am looking for

• the analogue of the prime decomposition and of the torus decomposition for non-orientable closed three-dimensional manifolds;
• some general information about families of known examples (Seifert manifolds, analogues of Haken manifolds, ...);
• the proof of a statement that can be found in the introduction of $\textit{Non-orientable 3-manifolds of complexity up to 7}$, G. Amendola, B. Martelli without reference: "Among the $8$ three-dimensional geometries, only $5$ have non-orientable representatives.".

Answer 1

1. The prime decomposition theorem is slightly different for non-orientable 3-manifolds. There is a version decomposing along 2-spheres described in Hempel’s book, but there is a subtle flaw in the statement corrected by Bruce Trace. A possible more natural decomposition is along projective planes and 2-spheres, but here the prime summands may be orbifolds with isolated orbifold points which are the cone over a projective plane. However I’m not sure if this is written down somewhere.

As pointed out by Moishe Kohan, the geometric decomposition for non-orientable 3-manifolds is described in the last section of Scott’s “The Geometries of 3-manifolds”. Unfortunately this doesn’t seem to have been proved in full generality yet.

2. Classes of examples of non-orientable 3-manifolds may be made by taking mapping tori either of non-orientable surfaces or of non-orientable mapping classes of orientable surfaces. Note that if $M$ is closed, aspherical, and non-orientable, then $b_1(M)>0$ and $M$ is Haken. Another simple way to construct non-orientable manifolds is to take a complement of a tubular neighborhood of a knot and attach the boundary to a torus by a double cover where the meridian wraps twice and the longitude once.

3. There are no closed non-orientable 3-manifolds modeled on spherical, Nil, or $\widetilde{SL_2(\mathbb{R})}$ geometries. Manifolds modeled on these geometries are Seifert fibered with non-triviald Seifert bundle. If a Seifert fibered manifold is non-orientable, then the Euler class of the Seifert bundle of the two-fold orientable cover is zero (since there is an orientation-reversing involution either reversing the orientation of the fibers or the base), so the manifold cannot be modeled on one of these geometries.

Answer 2

There is a (slightly) less abstract proof that the “twisted Seifert geometries” (elliptic, Nil, and PSL) have no non-orientable manifold quotients. First show that orientation-reversing isometries of $S^3$ have fixed points. Next show that all isometries of Nil and PSL preserve orientation. This is discussed in Peter Scott’s article “The geometries of 3-manifolds”.