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Non-orientable real algebraic three-dimensional manifolds

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.".
Yromed
  • 183
  • 6