Configurations of surfaces in 4-manifolds

We consider collections of surfaces {F'¡} smoothly embedded, except for a finite number of isolated singularities, self-intersections, and mutual intersections, in a 4-manifold M. A small 3-sphere about each exceptional point will intersect these surfaces in a link. If [F¡] G H2(M) are linearly dependent modulo a prime power, we find lower bounds for 2 genus (/)■) in terms of the [F¡], and invariants of the links that describe the exceptional points. 0. Introduction. The following special case of our main theorem is easy to state. Theorem 0.1. Let M be a closed smooth 4-manifold and {F¡) a collection of n smoothly embedded surfaces in general position. Let x¡ = [F¡] G H2(M). Suppose U F¡ is connected and 2 a,*, = pry where p is a prime, 0 <a¡ <pr, and a¡ ¥= 0 mod p. Let # be the total number of intersection points. Then # +22 genus(^) > 2y(2 xi y) 2 XfXj sign M • <J + 2(n 1) dim H2(M, Z ). For example, according to a theorem of C. T. C. Wall [W] if M is a smooth closed simply-connected 4-manifold with indefinite quadratic form, then in M # S2 X S2 any primitive noncharacteristic class may be represented by an embedded 2-sphere. Let M be S2 X S2 and let Fx be a 2-sphere representing (0, 1, 0, 0) G H2(S2 X S2#S2 X S2) (with respect to the natural basis). Let F2 be a 2-sphere representing (a, b, 0, 0) transverse to Fx where a > 1, b > 0 and (a, b) = 1. Let # be the total number of intersection points of Fx and F2. Then we have

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