EMBEDDINGS AND IMMERSIONS OF A 2-SPHERE IN 4-MANIFOLDS

Let M be Cp2#(-CP2)#P1# ... #P(m+k), where P1, ... , P(m+k) are copies of -Cp2. Let h, g, g1, ... , g(m+k) be the images of the standard gen erators of H2(CP2; Z), H-2(-CP2; Z), H-2(P1; Z), ... , H-2(P(m+k); Z) in H-2(M ; Z) respectively. Let xi = ph + qg + SIGMA(i=1)m r(i)g(i) be an element of H-2(M ; Z). Suppose xi2 = l > 0, p2 - q2 > 8, \p\ - \q\ greater-than-or-equal-to 2, and r(i) not-equal 0, i = 1, ... , m. If 2(m + 1 - 2) greater-than-or-equal-to p2 - q2, then xi cannot be repre sented by a smoothly embedded 2-sphere. If 2(m + r +[(l-r-1)/4]-1) gre ater-than-or-equal-to p2 - q2 for some r with 0 less-than-or-equal-to r less-than-or-equal-to l-1, then for a normal immersion f of a 2-sphe re representing xi the number of points of positive self-intersection d(f) greater-than-or-equal-to [(l-r-1)/4] + 1.