In dimension 2, there are two remarkable non-orientable closed manifolds, the projective plane (from synthetic geometry; has the fixed point property; algebraic compactification of the plane etc) and the Klein bottle (nowhere vanishing vector field; with immersions sold in your nearest nonorientable store). There is also a classification of all closed non-orientable surfaces, as connected sums of projective planes.
I am looking for examples of non-orientable 3 dimensional closed (compact, boundaryless) manifolds. Any with some special properties or arising from interesting geometrical problems? Is there a simple classification for them?
Best Answer
One gets non-orientable closed 3-manifolds by taking a non-orientable surface, and crossing with $S^1$, such as $P^2\times S^1$.
In fact, the geometrization theorem hasn't been proven completely for non-orientable 3-manifolds. Of course, a 2-fold cover has a connect sum and geometric decomposition. But the problem is that no one has shown that this decomposition can be made equivariant with respect to the covering translation. One expects that a careful analysis of the proof using Ricci flow could be made equivariant (at least Ricci flow preserves symmetries). One can't do the initial connect sum decomposition, because 1-sided projective planes must be cut along and coned off, resulting in an orbifold with isolated cone points.
In any case, I think that the most interesting non-orientable 3-manifold is the smallest known volume closed manifold, which has volume $2.0298...$, the same as the figure 8 complement, and fibers over the circle. This was discovered by Weeks in his census. I think it is known to be arithmetic. See Table 2 of a paper by Hodgson and Weeks.