Disclaimer - I don't have much experience in topology/complex geometry, so I apologize if what I'm asking is too elementary for this site.


Let $S$ be some orientable surface obtained by removing finitely many points from a closed surface.


The Riemann moduli space for $S$ is just the set of isomorphism classes of complex structures. In other words, it's the quotient of the set of all complex structures on $S$, by the equivalence relation where two complex structures $\mu,\mu'$ on $S$ are equivalent if there is an homeomorphism $f : S\rightarrow S$ such that $\mu\circ f = \mu'$.


According to Wikipedia, the Teichmuller space for $S$ can be described as also the quotient of the set of all complex structures on $S$ by the same equivalence relation, except where the $f$ is required to be isotopic to the identity.


In Lochak's paper here: <http://www.math.jussieu.fr/~lochak/textes/curves.pdf>


he says that points of the Teichmuller space can be described by pairs $(X,f)$ where $X$ is a Riemann surface (ie, a complex structure on $S$), and $f : S\rightarrow X$ is a diffeomorphism, and that $(X,f)\sim(X',f')$ if $f'\circ f^{-1}$ is homotopic to an isomorphism (ie, a biholomorphic map) from $X\rightarrow X'$.


My first question is - In this case, is $f'\circ f^{-1}$ being homotopic to an isomorphism equivalent to it being isotopic to one?


My second question is - Why are these two definitions equivalent? I can't seem to see why having an isomorphism $X\rightarrow X'$ that is isotopic to the identity is equivalent to $f'\circ f^{-1}$ being homotopic to an isomorphism.


In fact this just seems false. Ie, consider the case where $X = X'$ as Riemann surfaces, and pick markings $f,f'$ such that $f'\circ f^{-1}$ is not isotopic to the identity, then the equivalence of the two definitions would seem to imply that there are isomorphisms between $X$ and $X'$ that are isotopic to the identity, and ones that are not, which seems strange to me.



Elementary question about Isotopy (in the definition of a Teichmuller space)
1. D.B.A. Epstein proved in his paper "Curves on 2-manifolds and isotopies", Acta Math. 1966, that homotopy implies isotopy for homeomorphisms of surfaces of finite type.
2. There are conformal automorphisms of Riemann surfaces which are not isotopic to the identity and there is the trivial automorphism, which is the identity (you can aso use rotations of tori as trivial examples). It is a corollary of Hurwitz theorem that a conformal automorphism of a hyperbolic surface is homotopic to identity iff it is the identity. The easiest examples of nontrivial automorphisms are hyperelliptic involutions, but there are many more. Just think about covering transformations of a finite cover $S'\to S$.


