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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 $Ad\phi$is 6g-6 dimension exactly from differential form tensor sections in bundles.

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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# Get $H^1(S,sl(2,R)_{Ad\phi}$) dimension directly from differential forms

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 $Ad\phi$ is 6g-6 dimension exactly from differential form tensor sections in bundles.

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.