  In a previous paper we have constructed an invariant of four-dimensional
manifolds with boundary in the form of an element in the stable homotopy group
of the Seiberg-Witten Floer spectrum of the boundary. Here we prove that when
one glues two four-manifolds along their boundaries, the Bauer-Furuta invariant
of the resulting manifold is obtained by applying a natural pairing to the
invariants of the pieces. As an application, we show that the connected sum of
three copies of the K3 surface contains no exotic nuclei. In the process we
also compute the Floer spectrum for several Seifert fibrations.
