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Most 3-manifold topologists tend to hypothesize away 2-sided projective planes. If a 3-manifold contains a 2-sided projective plane, then it must be non-orientable, and the preimage of the projective plane in the orientable double cover must be essential. Cutting along a maximal collection of disjoint essential embedded 2-spheres in the orientable cover (which may be made equivariant by Meeks-Yau), and capping off with spheres, one obtains an irreducible 3-manifold with an orientation reversing involution. Corresponding to projective planes will be spheres coned off which are acted on by the involution as the suspension on the antipodal map. If the components of this 3-manifold are atoroidal, then this is covered by a result of Dinkelbach and Leeb for spherical, hyperbolic, and $S^2\times R$ metrics. The case of the other homogeneous metrics was taken care of before by Meeks and Scott.

As far as I can tell, the toroidal case may still be open in general. One may argue that the involution preserves the JSJ tori, so it preserves the decomposition into geometric pieces. But I'm not sure it is now completely proven that the action on each geometric piece is standard, so that the quotient is geometric on each JSJ piece, because the results quoted above only cover involutions on closed geometric manifolds. A theorem of Hatcher implies that it is isotopic to such an involution, but I'm not sure his theorem implies that the isotopy is achieved by a path of involutions (with fixed points).

Kleiner and Lott are working on a proof of the orbifold theorem using Ricci flow, and I think their argument ought to give a unified argument for all of these things.

For aspherical non-orientable 3-manifolds, I think the proof via Ricci flow still works, even though Perelman didn't formulate it this way. Or Thurston's original proof should work, since these are Haken manifolds.

Most 3-manifold topologists tend to hypothesize away 2-sided projective planes. If a 3-manifold contains a 2-sided projective plane, then it must be non-orientable, and the preimage of the projective plane in the orientable double cover must be essential. Cutting along a maximal collection of disjoint essential embedded 2-spheres in the orientable cover (which may be made equivariant by Meeks-Yau), and capping off with spheres, one obtains an irreducible 3-manifold with an orientation reversing involution. Corresponding to projective planes will be spheres coned off which are acted on by the involution as the suspension on the antipodal map. If the components of this 3-manifold are atoroidal, then this is covered by a result of Dinkelbach and Leeb for spherical, hyperbolic, and $S^2\times R$ metrics. The case of the other homogeneous metrics was taken care of before by Meeks and Scott.

As far as I can tell, the toroidal case may still be open in general. One may argue that the involution preserves the JSJ tori, so it preserves the decomposition into geometric pieces. But I'm not sure it is now completely proven that the action on each geometric piece is standard, so that the quotient is geometric on each JSJ piece, because the results quoted above only cover involutions on closed geometric manifolds. A theorem of Hatcher implies that it is isotopic to such an involution, but I'm not sure his theorem implies that the isotopy is achieved by a path of involutions (with fixed points).

Kleiner and Lott are working on a proof of the orbifold theorem using Ricci flow, and I think their argument ought to give a unified argument for all of these things.

For aspherical non-orientable 3-manifolds, I think the proof via Ricci flow still works, even though Perelman didn't formulate it this way. Or Thurston's original proof should work, since these are Haken manifolds.

Most 3-manifold topologists tend to hypothesize away 2-sided projective planes. If a 3-manifold contains a 2-sided projective plane, then it must be non-orientable, and the preimage of the projective plane in the orientable double cover must be essential. Cutting along a maximal collection of disjoint essential embedded 2-spheres in the orientable cover (which may be made equivariant by Meeks-Yau), and capping off with spheres, one obtains an irreducible 3-manifold with an orientation reversing involution. Corresponding to projective planes will be spheres coned off which are acted on by the involution as the suspension on the antipodal map. If the components of this 3-manifold are atoroidal, then this is covered by a result of Dinkelbach and Leeb for spherical, hyperbolic, and $S^2\times R$ metrics. The case of the other homogeneous metrics was taken care of before by Meeks and Scott.

As far as I can tell, the toroidal case may still be open in general. One may argue that the involution preserves the JSJ tori, so it preserves the decomposition into geometric pieces. But I'm not sure it is now completely proven that the action on each geometric piece is standard, so that the quotient is geometric on each JSJ piece, because the results quoted above only cover involutions on closed geometric manifolds. A theorem of Hatcher implies that it is isotopic to such an involution, but I'm not sure his theorem implies that the isotopy is achieved by a path of involutions (with fixed points).

Kleiner and Lott are working on a proof of the orbifold theorem using Ricci flow, and I think their argument ought to give a unified argument for all of these things.

For aspherical non-orientable 3-manifolds, I think the proof via Ricci flow still works, even though Perelman didn't formulate it this way. Or Thurston's original proof should work, since these are Haken manifolds.

Most 3-manifold topologists tend to hypothesize away 2-sided projective planes. If a 3-manifold contains a 2-sided projective plane, then it must be non-orientable, and the preimage of the projective plane in the orientable double cover must be essential. Cutting along a maximal collection of disjoint essential embedded 2-spheres in the orientable cover (which may be made equivariant by Meeks-Yau), and capping off with spheres, one obtains an irreducible 3-manifold with an orientation reversing involution. Corresponding to projective planes will be spheres coned off which are acted on by the involution as the suspension on the antipodal map. If the components of this 3-manifold are atoroidal, then this is covered by a result of Dinkelbach and Leeb for spherical, hyperbolic, and $S^2\times R$ metrics. The case of the other homogeneous metrics was taken care of before by Meeks and Scott.

As far as I can tell, the toroidal case may still be open in general. One may argue that the involution preserves the JSJ tori, so it preserves the decomposition into geometric pieces. But I'm not sure it is now completely proven that the action on each geometric piece is standard, so that the quotient is geometric on each JSJ piece, because the results quoted above only cover involutions on closed geometric manifolds. A theorem of Hatcher implies that it is isotopic to such an involution, but I'm not sure his theorem implies that the isotopy is achieved by a path of involutions (with fixed points).

Kleiner and Lott are working on a proof of the orbifold theorem using Ricci flow, and I think their argument ought to give a unified argument for all of these things.

For aspherical non-orientable 3-manifolds, I think the proof via Ricci flow still works, even though Perelman didn't formulate it this way. Or Thurston's original proof should work, since these are Haken manifolds.

Most 3-manifold topologists tend to hypothesize away 2-sided projective planes. If a 3-manifold contains a 2-sided projective plane, then it must be non-orientable, and the preimage of the projective plane in the orientable double cover must be essential. Cutting along a maximal collection of disjoint essential embedded 2-spheres in the orientable cover (which may be made equivariant by Meeks-Yau), and capping off with spheres, one obtains an irreducible 3-manifold with an orientation reversing involution. Corresponding to projective planes will be spheres coned off which are acted on by the involution as the suspension on the antipodal map. If the components of this 3-manifold are atoroidal, then this is covered by a result of Dinkelbach and Leeb for spherical, hyperbolic, and $S^2\times R$ metrics. The case of the other homogeneous metrics was taken care of before by Meeks and Scott.

As far as I can tell, the toroidal case may still be open in general. One may argue that the involution preserves the JSJ tori, so it preserves the decomposition into geometric pieces. But I'm not sure it is now completely proven that the action on each geometric piece is standard, so that the quotient is geometric on each JSJ piece, because the results quoted above only cover involutions on close geometric manifolds. A theorem of Hatcher implies that it is isotopic to such an involution, but I'm not sure his theorem implies that the isotopy is achieved by a path of involutions (with fixed points).

Kleiner and Lott are working on a proof of the orbifold theorem using Ricci flow, and I think their argument ought to give a unified argument for all of these things.

For aspherical non-orientable 3-manifolds, I think the proof via Ricci flow still works, even though Perelman didn't formulate it this way. Or Thurston's original proof should work, since these are Haken manifolds.

Most 3-manifold topologists tend to hypothesize away 2-sided projective planes. If a 3-manifold contains a 2-sided projective plane, then it must be non-orientable, and the preimage of the projective plane in the orientable double cover must be essential. Cutting along a maximal collection of disjoint essential embedded 2-spheres in the orientable cover (which may be made equivariant by Meeks-Yau), and capping off with spheres, one obtains an irreducible 3-manifold with an orientation reversing involution. Corresponding to projective planes will be spheres coned off which are acted on by the involution as the suspension on the antipodal map. If the components of this 3-manifold are atoroidal, then this is covered by a result of Dinkelbach and Leeb for spherical, hyperbolic, and $S^2\times R$ metrics. The case of the other homogeneous metrics was taken care of before by Meeks and Scott.

As far as I can tell, the toroidal case may still be open in general. One may argue that the involution preserves the JSJ tori, so it preserves the decomposition into geometric pieces. But I'm not sure it is now completely proven that the action on each geometric piece is standard, so that the quotient is geometric on each JSJ piece, because the results quoted above only cover involutions on closed geometric manifolds. A theorem of Hatcher implies that it is isotopic to such an involution, but I'm not sure his theorem implies that the isotopy is achieved by a path of involutions (with fixed points).

Kleiner and Lott are working on a proof of the orbifold theorem using Ricci flow, and I think their argument ought to give a unified argument for all of these things.

For aspherical non-orientable 3-manifolds, I think the proof via Ricci flow still works, even though Perelman didn't formulate it this way. Or Thurston's original proof should work, since these are Haken manifolds.