Extended Regge Complex for Linearized Riemann-Cartan Geometry and Cohomology

Abstract We show that the cohomology of the Regge complex in three dimensions is isomorphic to $$ \mathcal {H}^{{\scriptscriptstyle \bullet }}_{dR}(\varOmega )\otimes \mathcal{R}\mathcal{M}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>dR</mml:mi> </mml:mrow> <mml:mstyle> <mml:mo>∙</mml:mo> </mml:mstyle> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⊗</mml:mo> <mml:mi>R</mml:mi> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> , the de Rham cohomology of differential forms with values in infinitesimal rigid body motions. Based on an observation that the twisted de Rham complex extends the elasticity complex (based on Riemannian deformation) 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.