  A localisation of the category of n-manifolds is introduced by formally
inverting the connected sum construction with a chosen n-manifold Y. On the
level of automorphism groups, this leads to the stable diffeomorphism groups of
n-manifolds. In dimensions 0 and 2, this is connected to the stable homotopy
groups of spheres and the stable mapping class groups of Riemann surfaces. In
dimension 4 there are many essentially different candidates for the n-manifold
Y to choose from. It is shown that the Bauer--Furuta invariants provide
invariants in the case Y = CP^2, which is related to the birational
classification of complex surfaces. This will be the case for other Y only
after localisation of the target category. In this context, it is shown that
the K3-stable Bauer--Furuta invariants determine the S^2xS^2-stable invariants.
