  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 Weitzenbock connection, we have extracted the torsion invariant T_G,
equivalent (up to boundary terms) to the Gauss-Bonnet term G. T_G 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
Weitzenbock connection, T_G depends only on the vielbein, and this allows us to
consider a novel class of modified gravity theories based on F(T,T_G), 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,T_G) gravity.
