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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:

  • Is every $ \mathbb{RP}^n $ bundle over $ S^1 $ trivial?

  • 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).

Ian Gershon Teixeira
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  • All fiber bundles over spheres should arise from the clutching construction. I think this should give you a way to classify the fiber bundles. – Jeroen van der Meer Jan 05 '22 at 14:19
  • @JeroenvanderMeer Correct, they are asking about the mapping class group of $\mathbb{R}P^n$ for the first question which I presume is written about somewhere. The second question is easier and can be dealt with just by cohomological means using the fact that $\mathbb{R}P^\infty$ represents first $\mathbb{Z}/2$ cohomology. – Connor Malin Jan 05 '22 at 16:35
  • You mean: $\mathrm{Diff}(\Bbb RP^n)=\begin{cases}Id& n=2k\\Bbb Z_2& n=2k+1\end{cases}$? – C.F.G Jan 10 '22 at 05:07
  • No I mean every diffeomorphism is homotopic to one of those options as stated in my question. For the definition of homotopy see https://en.m.wikipedia.org/wiki/Homotopy Indeed the statement you suggest in your comment is certainly false for $ n \geq 1$ since any manifold of dimension $ n \geq 1 $ has an infinite dimensional diffeomorphism group. Your statement happens to be true for $n=0 $ because then the manifold is a single point and the diffeomorphism group of a 0 dimensional manifold is just the group of all permutations of the points of the manifold. – Ian Gershon Teixeira Jan 11 '22 at 04:02
  • Do you really want "homotopy"? "Isotopy" seems more natural to consider in this context. (But I don't know the answer either way!) Also, in the first question, what structure group are you considering? Based on your statement about $S^n$, it seems you are consider only linear $S^n$ bundles over $S^1$ (i.e., the structure group is a subgroup of the general linear group.) – Jason DeVito - on hiatus Jan 18 '22 at 16:23
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    Apparently the mapping class group of $\mathbb CP^n$ is not known, and I suspect $\mathbb RP^n$ is a harder question. – Cheerful Parsnip Jan 22 '22 at 05:20

1 Answers1

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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.

Ian Gershon Teixeira
  • 10,176