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Let 
$\Gamma \subset SO(3,1)$ be a lattice. The well known bending deformations, introduced by Thurston and Apanasov, can be used to construct non-trivial curves of representations of 
$\Gamma $ into 
$SO(4,1)$ when 
$\Gamma \backslash {{\mathbb{H}}^{3}}$ contains an embedded totally geodesic surface. A tangent vector to such a curve is given by a non-zero group cohomology class in 
${{\text{H}}^{1}}\left( \Gamma ,\mathbb{R}_{1}^{4} \right)$ . Our main result generalizes this construction of cohomology to the context of “branched” totally geodesic surfaces. We also consider a natural generalization of the famous cuspidal cohomology problem for the Bianchi groups (to coefficients in non-trivial representations), and perform calculations in a finite range. These calculations lead directly to an interesting example of a link complement in 
${{S}^{3}}$ which is not infinitesimally rigid in $SO(4,1)$ . The first order deformations of this link complement are supported on a piecewise totally geodesic 2-complex.
