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For a genus g surface S with fundamental group $\pi$, consider Teichmuller space $Hom(\pi,SL(2,R))/SL(2,R)$, we identify tangent space at point $\phi\in Hom(\pi,SL(2,R))/SL(2,R)$ as $H^1(S,sl(2,R)_{Ad\phi})$,

where $sl(2,R)_{Ad\phi}$ is the flat Lie algebra bundle on surface with holonomy representation of $ Ad\phi$.

I am wondering how to easily get $H^1(S,sl(2,R)_{Ad\phi}$) is 6g-6 dimension directly from differential forms, not passing through group cohomology or Serre duality. For me, it is more likely to be 6g dimesion.

I will be also appreciated if someone tell me some reference books for twisted cohomology in differential forms. Thanks.

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  • $\begingroup$ Actually, Teichmuller space naturally identifies with a certain component of $Hom(π,SL(2,{\mathbb R}))/SL(2,{\mathbb R})$ consisting of discrete and faithful representations. Then the simplest computation of $H^1$ is combination of Poincare duality and Euler characteristic: $\chi(S, {\mathbb V})= dim(V) \chi(S)$, where ${\mathbb V}$ the the flat bundle over $S$ associated with the surface group action on $V=psl(2,{\mathbb R})$. You also have $0=H^0\cong H^2$ by Poincare duality and the fact that centralizer of $\phi(\pi)$ is trivial. Thus, $dim H^1= (2g-2) dim(V)= 6g-6$. $\endgroup$
    – Misha
    Commented Apr 19, 2012 at 0:18
  • $\begingroup$ Dr.Misha, thank you, that gives me hint to find the right elements in $H^1$. $\endgroup$
    – Qiongling Li
    Commented Apr 19, 2012 at 3:45
  • $\begingroup$ A good reference for differential forms with twisted coefficients is Raghunathan's book "Discrete subgroups of Lie groups." $\endgroup$
    – Misha
    Commented Apr 19, 2012 at 4:21
  • $\begingroup$ Dr.Misha, that is really a nice reference. Thank you. $\endgroup$
    – Qiongling Li
    Commented Apr 19, 2012 at 14:57

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