In this chapter we discuss Künneth geometry in real dimension four. Since in dimension two Künneth geometry is essentially Lorentz geometry, dimension four is really the first interesting case. For at least two reasons, it is also a very special case. First, it is possible to classify almost Künneth structures in terms of classical invariants. Second, four-dimensional symplectic geometry is very subtle, and symplectic structures in this dimension are constrained by their relation with Seiberg-Witten gauge theory. We will see that this makes it likely that Künneth four-manifolds may be classified, although we do not achieve that goal here, except in the hypersymplectic case.

Throughout this chapter we will use not only the material developed in earlier chapters of this book, but also the tools of modern four-dimensional geometry and topology. In particular we will use results from gauge theory. A good reference for both the basics of four-dimensional differential topology and results from Donaldson theory is the book by Donaldson and Kronheimer [DK-90]. In fact, very little Donaldson theory will be used in this chapter. We will make more use of results from Seiberg-Witten theory, for which we refer to the book by Morgan [Mor-96] and the second author’s Bourbaki lecture [Kot-97a] on Taubes’s work.