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.

翻译：我们证明了三维Regge复形的上同调同构于$\mathcal{H}^{{\scriptscriptstyle \bullet}}_{dR}(\Omega)\otimes\mathcal{RM}$，即无穷小刚体运动取值的de~Rham上同调。基于对扭曲de~Rham复形将弹性（Riemann变形）复形扩展至线性化余框架、联络1-形式、曲率与Cartan挠率的观察，我们在任意三角剖分上构建了线性化Riemann-Cartan几何的离散版本，并确定了其上的上同调。