  We present a method to desingularize a compact G_2 manifold with isolated
conical singularities by cutting out a neighbourhood of each singular point and
glueing in an asymptotically conical G_2 manifold. Controlling the error on the
overlap glueing region enables us to use a result of Joyce to conclude that the
resulting compact smooth 7-manifold admits a torsion-free G_2 structure, with
full G_2 holonomy.
  There are topological obstructions for this procedure to work, which arise
from the degree 3 and degree 4 cohomology of the asymptotically conical G_2
manifolds which are glued in at each conical singularity. When a certain
necessary topological condition on the manifold with isolated conical
singularities is satisfied, we can introduce correction terms to the glueing
procedure to ensure that it still works. In the case of degree 4 obstructions,
these correction terms are trivial to construct, but in the case of degree 3
obstructions we need to solve an elliptic equation on a non-compact manifold.
For this we use the Lockhart-McOwen theory of weighted Sobolev spaces on
manifolds with ends. This theory is also used to obtain a good asymptotic
expansion of the G_2 structure on an asymptotically conical G_2 manifold under
an appropriate gauge-fixing condition, which is required to make the glueing
procedure work.
