closed geodesic $\gamma$ on $M$. We have to show that the cohomology class $\Theta(f)$ does not vanish. Using the exact sequence (4) above, this is the case if and only if a quasi-morphism $\Psi(f)$ representing $\Theta(f)$ is not equivalent to any morphism for $\Gamma$.

For this we argue as before. Namely, let $\rho:\Gamma\to \mathbb{R}$ be any morphism for $\Gamma$ and let $\beta$ be a smooth closed 1-form on $M$ defining $\rho$. By assumption, there is a periodic point $v\in L$ of period $\tau>0$ for the geodesic flow $\Phi^t$ such that $\int_0^\tau
(f-\beta)(\Phi^tv)dt=c>0$. Let $\tilde v$ be a lift of $v$ to $\tilde L$ and let $p\in \tilde M$ be the foot-point of $\tilde
v$. Choose an extension of $f$ to a locally Hölder continuous function $F$ on $T^1M$ and let $\tilde F$ be the lift of $F$ to $T^1\tilde
M$. By definition, the quasi-morphism $\Psi(f)$ induced by $F$ and the choice of the basepoint $p$ assigns to $g\in \Gamma$ the integral $\int_0^T \tilde F(\gamma_g^\prime(s))ds$ where $\gamma_g:[0,T]\to
\tilde M$ is the oriented geodesic arc connecting $p$ to $g(p)$. Moreover, this quasi-morphism represents the class $\Theta(f)$. Now let $\eta$ be the geodesic in $\tilde M$ which is tangent to $\tilde v$. By 