  Two 4-manifolds are stably diffeomorphic if they become diffeomorphic after
connected sum with S^2 x S^2's. This paper shows that two closed, orientable,
homotopy equivalent, smooth 4-manifolds are stably diffeomorphic, provided a
certain map from the second homology of the fundamental group with coefficients
in Z/2 to the L-theory of the group is injective. This injectivity is implied
by the Borel/Novikov conjecture for torsion-free groups, which is known for
many groups. There are also results concerning the homotopy invariance of the
Kirby-Siebenmann invariant. The method of proof is to use Poincare duality in
Spin bordism to translate between Wall's classical surgery and Kreck's modified
surgery.
