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 Weitzenböck 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 Weitzenböck 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.

We investigate the cosmological applications of $F(T,T_G)$ gravity, which is a novel modified gravitational theory based on the torsion invariant $T$ and the teleparallel equivalent of the Gauss-Bonnet term $T_G$. $F(T,T_G)$ gravity differs from both $F(T)$ theories as well as from $F(R,G)$ class of curvature modified gravity, and thus its corresponding cosmology proves to be very interesting. In particular, it provides a unified description of the cosmological history from early-times inflation to late-times self-acceleration, without the inclusion of a cosmological constant. Moreover, the dark energy equation-of-state parameter can be quintessence or phantomlike, or experience the phantom-divide crossing, depending on the parameters of the model.

We study spherically symmetric solutions in scalar-torsion gravity theories in which a scalar field is coupled to torsion with a derivative coupling. We obtain the general field equations from which we extract a decoupled master equation, the solution of which leads to the specification of all other unknown functions. We first obtain an exact solution which represents a new wormholelike solution dressed with a regular scalar field. Then, we find large distance linearized spherically symmetric solutions in which the space asymptotically is anti–de Sitter.