In the comments to Mapping torus of orientation reversing isometry of the sphere it was stated that there are only two $ S^n $ bundles over $ S^1 $ up to diffeomorphism. The conversation related to this led me to wonder several things:
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Is every $ \mathbb{RP}^n $ bundle over $ S^1 $ trivial?
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Every diffeomorphism of the sphere is either homotopic to the identity or to an orientation reversing isometry. Is every diffeomorphism of even dimensional real projective space homotopic to the identity and every diffeomorphism of odd dimensional projective space is homotopic to either the identity or to an orientation reversing isometry?
I expect the answer to my first question is yes for even $ n $ and no for odd $ n $. Basically because there are exactly 2 sphere bundles over the circle one the mapping torus of orientation preserving maps (the trivial bundle) and one for orientation reversing maps (the non trivial bundle). So importing that intuition to the case of $ RP^n $ then the orientable $ RP^n $ should have two bundles over the circle and the non orientable should have just one. For $ n=1 $ this checks out since that projective space is orientable and thus we have exactly two bundles over the circle (the trivial one/ the 2 torus and the nontrivial one/ the Klein bottle).
Best Answer
My second question is unambiguous and can be answered affirmatively, as Connor Malin suggested in his comment, by cohomological means using $ \mathbb{R}P^\infty $. This is done by Dmitry Vaintrob in his answer here
https://mathoverflow.net/questions/414465/mathbbrpn-bundles-over-the-circle/414493?noredirect=1#comment1062832_414493
Furthermore, Dimtry Vaintrob notes that this result answers my first question up to homotopy and the classification is as I conjectured. However Tom Goodwillie notes in his answer to the same question that the classification is wild up to diffeomorphism because of exotic smooth spheres and therefore my conjecture about $ \mathbb{R}P^n \rtimes S^1 $ is false working up to diffeomorphism.
Indeed, a big moral of the story for me with this experience has been that my question "Classify $ \mathbb{R}P^n $ bundles over $ S^1 $" is very vague since topologists use many notions of equivalence. Including classification up to homotopy, or homeomorphism, or diffeomorphism, or even classification as bundles with a certain structure group.