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In "An introduction to Teichmüler Theory" of Yoichi Imayoshi and Masahiko Taniguchi the Teichmüller space is defined as follows: fix a compact Riemann surface $R$ of genus $g$, a marking on a Riemann surface $S$ (of genus $g$) is an orientation-preserving diffeomorphism $f:R\rightarrow S$. Two markings $(S,f)$ and $(S',f')$ are declared equivalent if and only if there is a biholomorphic map $h:S\rightarrow S'$ such that the map $g\circ h \circ f^{-1}:R\rightarrow R$ is homotopic to the identity. The equivalence classes give the genus $g$ Teichmuller space $T(R)$.

  • Is this definition assuming that between any two smooth surfaces there exists an orientation preserving diffeomorphism? Is this not a huge assumption? Would we loose any generality if instead of asking them to be diffeos we asked for homeos? (I don't think it matters, at least in section 1.2 and 1.3 of chapter 1).

EDIT: I understand that it is assuming the uniqueness of smooth structures mod diffeos. In any case, how does this relate to the notion of defining the Teichmuller space declaring two objects equivalent if they is a biholomorphic map that is is homotopic to the identity? (which seems much more intuitive).

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    $\begingroup$ 1) There certainly is an orientation-preserving diffeomorphism between closed surfaces of genus $g$; this is the classification of surfaces. 2) Homemomorphic smooth surfaces are diffeomorphic. This is an old, old theorem, and there is a nice writeup using newer language by Hatcher here. $\endgroup$
    – mme
    Sep 22, 2020 at 22:38
  • $\begingroup$ Hello, thank you very much. 1) I don't get how from the classification of surfaces we can always find an orientation preserving diffeo between any two closed surfaces. We can surely find a homeo and use it so the other surface inherits the smooth structure of the other, but "a priori" without point (2) this would not yield a diffeo for any two smooth structures. (2) I know of the fact, part of the question comes from that is seems to me that it's quite a big assumption to make. In any case, thank you and I will read the writeup. $\endgroup$
    – Gerard Bargalló
    Sep 23, 2020 at 7:47
  • $\begingroup$ "The classification of surfaces" means both the the topological and smooth classification. You can prove it smoothly either by largely the same arguments, taking care with the corners, or by way of Morse theory --- you get a handlebody decomposition and you can do handleslides and handle cancelations to put your decomposition into a standard form. $\endgroup$
    – mme
    Sep 23, 2020 at 10:16
  • $\begingroup$ Okay, thank you very much!! $\endgroup$
    – Gerard Bargalló
    Sep 24, 2020 at 18:48

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