  We study axially symmetric codimension-2 cosmology for a distributional
braneworld fueled by a localised 4D perfect fluid, in a 6D Lovelock theory. We
argue that only the matching conditions (dubbed topological) where the
extrinsic curvature on the brane has no jump describe a pure codimension-2
brane. If there is discontinuity in the extrinsic curvature on the brane, this
induces inevitably codimension-1 distributional terms. We study these
topological matching conditions, together with constraints from the bulk
equations evaluated at the brane position, for two cases of regularisation of
the codimension-2 defect. First, for an arbitrary smooth regularisation of the
defect and second for a ring regularisation which has a cusp in the angular
part of the metric. For a cosmological ansatz, we see that in the first case
the coupled system is not closed and requires input from the bulk equations
away from the brane. The relevant bulk function, which is a time-dependent
angular deficit, describes the energy exchange between the brane and the 6D
bulk. On the other hand, for the ring regularisation case, the system is closed
and there is no leakage of energy in the bulk. We demonstrate that the full set
of matching conditions and field equations evaluated at the brane position are
consistent, correcting some previous claim in the literature which used rather
restrictive assumptions for the form of geometrical quantities close to the
codimension-2 brane. We analyse the modified Friedmann equation and we see that
there are certain corrections coming from the non-zero extrinsic curvature on
the brane. We establish the presence of geometric self-acceleration and a
possible curvature domination wedged in between the period of matter and
self-acceleration eras as signatures of codimension-2 cosmology.
