  We show that the cohomology of the Regge complex in three dimensions is
isomorphic to $\mathcal{H}^{{\scriptscriptstyle
\bullet}}_{dR}(\Omega)\otimes\mathcal{RM}$, the
infinitesimal-rigid-body-motion-valued de~Rham cohomology. Based on an
observation that the twisted de~Rham complex extends the elasticity (Riemannian
deformation) complex to the linearized version of coframes, connection 1-forms,
curvature and Cartan's torsion, we construct a discrete version of linearized
Riemann-Cartan geometry on any triangulation and determine its cohomology.
