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.

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.".

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.".

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.

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 ?

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.

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.".