  In the paper \cite{wall_1}, C.T.C. Wall proved that two smooth closed simply
connected 4-manifolds which are homeomorphic are in fact stably diffeomorphic.
We prove a similar result which states that two smooth closed 4-manifolds
satisfying certain properties are stably diffeomorphic if and only if their
signatures agree. The manifolds in question are obtained by surgery on loops.
The methods we use are modified surgery of Kreck \cite{kreck} and Kirby
calculus.
