Stable Diffeomorphism Groups of~4-Manifolds

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~$Y=\overline{\CC P}^2$, 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~$S^2\!\times\!S^2$-stable invariants.

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Published 1 January 2008