Embeddings and Immersions of a 2-Sphere in 4-Manifolds

Abstract

Let M be CP 2 #(−CP 2 )#P 1 ... #P m+k , where P 1 ,..., P m+k are copies of −CP 2 . Let h, g, g 1 ,..., g m+k be the images of the standard generators of H 2 (CP 2 ; Z), H 2 (−CP 2 ; Z), H 2 (P 1 ; Z),..., H 2 (P m+k ; Z) in H 2 (M; Z) respectively. Let ξ = ph + qg + ∑ i=1 m r i g i be an element of H 2 (M; Z). Suppose ξ 2 = l > 0, p 2 − q 2 ≥ 8, |p| − |q| ≥ 2, and r i ¬= 0, i = 1,..., m. If 2(m + l − 2) ≥ p 2 − q 2 , then ξ cannot be represented by a smoothly embedded 2-sphere. If 2(m + r + [(l − 1)/4] − 1) ≥ p 2 − q 2 for some r with 0 ≤ r ≤ l − 1, then for a normal immersion f of a 2-sphere representing ξ the number of points of positive self-intersection d f ≥ [l − r − 1)/4] + 1