Tight contact structures on connected sums need not be contact connected sums

In dimension three, convex surface theory implies that every tight contact structure on a connected sum M # N can be constructed as a connected sum of tight contact structures on M and N. I will explain some examples showing that this is not true in any dimension greater than three.  The proof is based on a recent higher-dimensional version of a classic result of Eliashberg about the symplectic fillings of contact manifolds obtained by subcritical surgery. This is joint work with Paolo Ghiggini and Klaus Niederkrüger.