THE MINIMAL GENUS PROBLEM IN RULED MANIFOLDS

In this paper, we consider the minimal genus problem in a ruled 4-manifold M. There are three key ingredients in the studying, the action of diffeomorphism group of M on H2(M,Z), the geometric construction of surfaces representing a cohomology class and the generalized adjunction formula. At first, we discuss the standard form (see Definition 1.1) of a class under the action of diffeomorphism group on H2(M,Z), we prove the uniqueness of the standard form. Then we construct some embedded surfaces representing the standard forms of some positive classes, the generalized adjunction formula is used to show that these surfaces realize the minimal genera.

FOUR-DIMENSIONAL INVARIANTS OF LINKS AND THE ADJUNCTION FORMULA

We compute the slice euler characteristic of certain links by using the so-called generalized adjunction formula, which is proved by Kronheimer, Mrowka, Morgan, Szabó and Taubes. Furthermore, for links obtained from such links by band surgery, we estimate the unknotting numbers, the 4-dimensional clasp numbers, and the slice euler characteristic.

Configurations of 2-spheres in the K3 surface and other 4-manifolds

A current theme in the theory of 4-manifolds is the study of which properties of complex surface are determined the underlying smooth 4-manifold. For instance, the genus of a complex curve in a complex surface is determined by its homology class, via the adjunction formula. Recent work in gauge theory [10–12] has shown that, to a great degree, a similar principal holds for an arbitrary (i.e. not necessarily complex) smooth representative of a 2-dimensional homology class. Another question, still unsolved even in the context of algebraic geometry, is to find the number of disjoint rational curves on a complex surface. The classical case, namely that of hypersurfaces in CP3, has only been settled for degrees d ≤ 6. The papers [1, 2, 4, 8, 14, 15] contain bounds on the number of such curves and constructions of surfaces with many ( — 2)-curves; the last two together establish that 65 is the correct bound in degree 6.

Embedded surfaces in four-manifolds, branched covers, and SO(3)-invariants

One of the outstanding problems in four-dimensional topology is to find the minimal genus of an oriented smoothly embedded surface representing a given homology class in a smooth four-manifold. For an arbitrary homology class in an arbitrary smooth manifold not even a conjectural lower bound is known. However, for the classes represented by smooth algebraic curves in (simply connected) algebraic surfaces, it is possible that the genus of the algebraic curve, given by the adjunction formulais the minimal genus. This is usually called the (generalized) Thom conjecture. It is mentioned in Kirby's problem list [11] as Problem 4·36.