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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 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.

3. There are no closed non-orientable 3-manifolds modeled on spherical, Nil, or $\widetilde{PSL_2(\mathbb{R})}$ geometries. Such manifolds are Seifert fibered with non-triviald Seifert bundle. If the manifold is non-orientable, then the Euler class of the bundle is zero so the manifold cannot be modeled on one of these geometries.

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.

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 Moisha 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 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.

3. There are no closed non-orientable 3-manifolds modeled on spherical, Nil, or $\widetilde{PSL_2(\mathbb{R})}$ geometries. Such manifolds are Seifert fibered with non-triviald Seifert bundle. If the manifold is non-orientable, then the Euler class of the bundle is zero so the manifold cannot be modeled on one of these geometries.

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 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.

3. There are no closed non-orientable 3-manifolds modeled on spherical, Nil, or $\widetilde{PSL_2(\mathbb{R})}$ geometries. Such manifolds are Seifert fibered with non-triviald Seifert bundle. If the manifold is non-orientable, then the Euler class of the bundle is zero so the manifold cannot be modeled on one of these geometries.

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.

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.

3. There are no closed non-orientable 3-manifolds modeled on spherical, Nil, or $\widetilde{PSL_2(\mathbb{R})}$ geometries. Such manifolds are Seifert fibered with non-triviald Seifert bundle. If the manifold is non-orientable, then the Euler class of the bundle is zero so the manifold cannot be modeled on one of these geometries.

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 Moisha 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 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.

3. There are no closed non-orientable 3-manifolds modeled on spherical, Nil, or $\widetilde{PSL_2(\mathbb{R})}$ geometries. Such manifolds are Seifert fibered with non-triviald Seifert bundle. If the manifold is non-orientable, then the Euler class of the bundle is zero so the manifold cannot be modeled on one of these geometries.