at.algebraic topology – $ mathbb{R}P^n $ bundles over the coterie Answer

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at.algebraic topology – $ mathbb{R}P^n $ bundles over the coterie

Is everybody $ mathbb{R}P^{2n} $ Bunch over the coterie trifling?

Is it precisely two? $ mathbb{R}P^{2n+1} $ bundles over the coterie?

This is a trial put up of (sever of) my MSE query

https://math.stackexchange.com/questions/4349052/diffeomorphisms-of-spheres-and-real-projective-spaces

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My instinct for considering each solutions are sure is that there are precisely 2 bundles of balls over the coterie. The trifling after which the non-trivial (and non-orientable) that may breathe realized as a mapping torus of an orientation-reversing mapping of the sphere. So signify this instinct into projective areas after which into the orientable $ mathbb{R}P^{2n+1} $ ought to have a non-trivial (and non-orientable) bundle over the coterie, whereas the non-orientable $ mathbb{R}P^{2n} $ ought to solely have the trifling bundle. for $n=1$ this holds as a result of this projective area is orientable and we thus have precisely two bundles over the coterie (the trifling = the 2-torus and the non-trivial = the Klein bottle).

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