  The usual Chern-Simons extension of Einstein gravity theory consists in
adding a squared Riemann contribution to the Hilbert Lagrangian, which means
that a square-curvature term is added to the linear-curvature leading term
governing the dynamics of the gravitational field. However, in such a way the
Lagrangian consists of two terms with a different number of curvatures, and
therefore not homogeneous. To develop a homogeneous Chern-Simons correction to
Einstein gravity we may, on the one hand, use the above-mentioned
square-curvature contribution as the correction for the most general
square-curvature Lagrangian, or on the other hand, find some linear-curvature
correction to the Hilbert Lagrangian. In the first case, we will present the
most general square-curvature leading term, which is in fact the already-known
re-normalizable Stelle Lagrangian. In the second case, the topological current
has to be an axial-vector built only in terms of gravitational degrees of
freedom and with a unitary mass dimension, and we will display such an object.
The comparison of the two theories will eventually be commented.
