Scalar curvature and the Thurston norm

the supremum being taken over all connected, oriented surfaces Σ embedded in Y whose genus g is at least 2 [8]. If Y contains spheres or non-separating tori, the definition is extended by declaring that α has infinite norm if it has non-zero pairing with any sphere or torus. It has been noted by various authors that there is a connection between the genus of embedded surfaces and solutions of the SeibergWitten monopole equations. In the present context, the result can be phrased as follows. The monopole equations depend on the choice of a metric h on Y and a Spinc structure c, to which we can associate the class c1(c), the first Chern class of the associated spin bundle. The result then states that, if the dual Thurston norm of c1(c) is greater than 1, then there exists a metric h on Y for which the corresponding monopole equations admit no solution. A fact that lies rather deeper is that the connection between the Thurston norm and the monopole equations is sharp, in the following sense, at least in the case that Y is irreducible (that is, every sphere bounds a ball in Y ). Let us say that a cohomology class α ∈ H2(Y ;R) is a monopole class if there is a Spinc structure c, with c1(c) = α over the reals, such that the corresponding monopole equations have a solution for all Riemannian metrics h on Y . Then we have: