Extended Regge complex for linearized Riemann-Cartan geometry and cohomology

We show that the cohomology of the Regge complex in three dimensions is isomorphic to ${\mathcal{H}}_{dR}^{\bullet }(\mathrm{\Omega })\otimes \mathcal{R}\mathcal{M}$, 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.