  This paper is concerned with the problem of stable diffeomorphism
classification of 4-manifolds obtained using the surgery on loops. The main
theorem states that under the assumption that the normal 1-type of two
4-manifolds in question is the same, the only classifying invariant is the
signature. In particular, in some cases, any two closed smooth 4-manifolds with
a given fundamental group, obtained by the standard construction, are stably
diffeomorphic.
