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When reading about this question which recently became active for some reason, I wanted to make a comment, as a warning regarding non-metrizable manifolds, to the effect that the every $C^\infty$ structure on the long (half-)line is compatible with at least continuum-many $C^\omega$ (i.e., real-analytic) structures, something which I seemed to remember, but I wasn't sure and couldn't remember a reference. The Wikipedia article on the long line states that “any given $C^\infty$ structure can be extended in infinitely many ways to different $C^\omega$ (=analytic) structures (which are pairwise non-diffeomorphic as analytic manifolds)”, but the reference given, viꝫ. the 1960 paper by H. Kneser & M. Kneser, “Reell-analytische Strukturen der Alexandroff-Halbgeraden und der Alexandroff-Geraden” (Archiv der Mathematik 11 104–106), as far as I can make it, only seems to support the weaker claim (Satz 5) that there are multiple analytic structures, not how many there are, nor any relation with a predetermined $C^\infty$ structure (the paper says nothing about $C^\infty$). The other paper cited in the Wikipedia article, by P. J. Nyikos, “Various smoothings of the long line and their tangent bundles”, Advances in Math. 93 (1992) 129–213, doesn't mention analytic structures at all except to note in the introduction that they can exist. So now I am completely confused as to what is known (perhaps as a folklore result with no precise reference). Specifically:

Question: Given $r<s$ in $\{0,1,2,\ldots,\infty,\omega\}$, how many different $C^s$ structures on the long line (or half-line) can exist that are compatible with a given $C^r$ structure? What are the most precise known results in this direction, and, specifically, is it correct that, as Wikipedia claims, for $r=\infty$ and $s=\omega$, every $C^\infty$ structure admits infinitely many different $C^\omega$ structures that are compatible with it? Are there other references than the two papers cited in Wikipedia (citations reproduced above)?

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    $\begingroup$ When you say "how many", you mean modulo what? Because if you push forward the $C^s$-structure by a non-$C^s$ $C^r$-diffeomorphism, you get another one. So you might mean "modulo $C^r$-diffeomorphism". $\endgroup$
    – YCor
    Sep 24, 2021 at 13:13
  • $\begingroup$ What I am asking is, given a $C^r$-diffeomorphism class $L$ of $C^r$ “manifolds” that have the long line as underlying topological space, what is the cardinality of the set of $C^s$-diffeomorphism classes of $C^s$ “manifolds” that have $L$ as underlying $C^r$ “manifold”. This number should be given in function of $r<s$ — and possibly $L$ if this matters. (So depending on what you mean by “modulo”, this is modulo $C^r$ or $C^s$. I think we've had this exact discussion nine months ago, and my position is still the same. 😉) $\endgroup$
    – Gro-Tsen
    Sep 24, 2021 at 19:36
  • $\begingroup$ Oh yes it amounts to the same: (1) $\mathrm{Diff}^r(X)$-orbits on the set of $C^s$-structures on $X$ extending the given $C^r$-structure, and (2) (all manifolds $C^r$-diffeomorphic to $X$) modulo $C^s$-diffeomorphism. $\endgroup$
    – YCor
    Sep 24, 2021 at 20:04

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