Teleparallel equivalent of Gauss-Bonnet gravity and its modifications

Inspired by the teleparallel formulation of general relativity, whose Lagrangian is the torsion invariant T, we have constructed the teleparallel equivalent of Gauss-Bonnet gravity in arbitrary dimensions. Without imposing the Weitzenb\"ock connection, we have extracted the torsion invariant T₆, equivalent (up to boundary terms) to the Gauss-Bonnet term G. T₆ is constructed by the vielbein and the connection, it contains quartic powers of the torsion tensor, it is diffeomorphism and Lorentz invariant, and in four dimensions it reduces to a topological invariant as expected. Imposing the Weitzenb\"ock connection, T₆ depends only on the vielbein, and this allows us to consider a novel class of modified gravity theories based on F (T, T₆), which is not spanned by the class of F (T) theories, nor by the F (R, G) class of curvature modified gravity. Finally, varying the action we extract the equations of motion for F (T, T₆) gravity.