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Often when people write about the geometrization conjecture they assume (for simplicity) that the manifold is orientable. I never seriously thought of non-orientable 3-manifolds, but while reading Morgan-Tian's paper I realized that they prove the geometrization for every compact 3-manifold $M$ with no embedded two-sided projective planes, which was news to me. There is a lot of notation in their paper so I ask

Question 1. Is the above a correct interpretation of what is proved in Morgan-Tian's paper?

Note: one way to rule out two-sided projective planes is to assume that $\pi_1(M)$ has no 2-torsion (because then a two-sided projective plane would lift to the orientation cover where it cannot exist). In particular, this gives the geometrization conjecture when $M$ is aspherical (in which case $\pi_1(M)$ is torsion free).

Question 2. What is the status of the geometrization conjecture for manifolds that contain two-sided projective planes?

Note: on the last two pages of Scott's wonderful survey "The Geometries of 3-manifolds" he describes a version of the geometrization conjecture that makes sense in the presence of two-sided projective planes.

UPDATE:

1. Looks like my reading of Morgan-Tian was hasty, and I no longer think they prove the geometrization for non-orientable manifolds that contain no two-sided projective planes. They only prove it for manifolds that become extinct in finite time under Ricci flow.

2. As we discussed with Ryan in comments the geometrization for manifolds that contain two-sided projective planes reduces to geometrization of certain non-orientable orbifolds with isolated singular points. However, in contrast with what Ryan says, I was unable to find this result the geometrization for such orbifolds in the literature. Again, lots of particular cases are known, but I could not find it claimed (let alone proved) in full generality.

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Often when people write about the geometrization conjecture they assume (for simplicity) that the manifold is orientable. I never seriously thought of non-orientable 3-manifolds, but while reading Morgan-Tian's paper I realized that they prove the geometrization for every compact 3-manifold $M$ with no embedded two-sided projective planes, which was news to me. There is a lot of notation in their paper so I ask

Question 1. Is the above a correct interpretation of what is proved in Morgan-Tian's paper?

Note: one way to rule out two-sided projective planes is to assume that $\pi_1(M)$ has no 2-torsion (because then a two-sided projective plane would lift to the orientation cover where it cannot exist). In particular, this gives the geometrization conjecture when $M$ is aspherical (in which case $\pi_1(M)$ is torsion free).

Question 2. What is the status of the geometrization conjecture for manifolds that contain two-sided projective planes?

Note: on the last two pages of Scott's wonderful survey "The Geometries of 3-manifolds" he describes a version of the geometrization conjecture that makes sense in the presence of two-sided projective planes.

UPDATE:

1. Looks like my reading of Morgan-Tian was hasty, and I no longer think they prove the geometrization for non-orientable manifolds that contain no two-sided projective planes. They only prove it for manifolds that become extinct in finite time under Ricci flow.

2. As we discussed with Ryan in comments the geometrization for manifolds that contain two-sided projective planes reduces to geometrization of certain non-orientable orbifolds with isolated singular points. However, in contrast with what Ryan says, I was unable to find this result in the literature. Again, lots of particular cases are known, but I could not find it claimed (let alone proved) in full generality.

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# Geometrization for 3-manifolds that contain two-sided projective planes

Often when people write about the geometrization conjecture they assume (for simplicity) that the manifold is orientable. I never seriously thought of non-orientable 3-manifolds, but while reading Morgan-Tian's paper I realized that they prove the geometrization for every compact 3-manifold $M$ with no embedded two-sided projective planes, which was news to me. There is a lot of notation in their paper so I ask

Question 1. Is the above a correct interpretation of what is proved in Morgan-Tian's paper?

Note: one way to rule out two-sided projective planes is to assume that $\pi_1(M)$ has no 2-torsion (because then a two-sided projective plane would lift to the orientation cover where it cannot exist). In particular, this gives the geometrization conjecture when $M$ is aspherical (in which case $\pi_1(M)$ is torsion free).

Question 2. What is the status of the geometrization conjecture for manifolds that contain two-sided projective planes?

Note: on the last two pages of Scott's wonderful survey "The Geometries of 3-manifolds" he describes a version of the geometrization conjecture that makes sense in the presence of two-sided projective planes.