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

• Published in: Proceedings of the American Mathematical Society 1993-08, Vol.118 (4), p.1323-1330
• Main Authors: Gan, Dan Yan, Guo, Jian Han
• Format: Article
• Language: English
• Subjects: Cauchy Schwarz inequality; Differential topology; Exact sciences and technology; Manifolds and cell complexes; Mathematical manifolds; Mathematical theorems; Mathematics; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/S0002-9939-1993-1152976-1

Let M be CP2#(-CP2) #P1# ⋯ #Pm + k, where P1, ..., Pm + kare copies of -CP2. Let h, g, g1, ..., gm + kbe the images of the standard generators of H2(CP2; Z), H2(-CP2; Z), H2(P1; Z), ..., H2(Pm + k; Z) in H2(M; Z) respectively. Let ξ = ph + qg + ∑m i = 1rig ibe an element of H2(M; Z). Suppose$\xi^2 = l > 0, p^2 - q^2 \geq 8, |p| - |q| \geq 2$, and ri≠ 0, i = 1, ..., m. If 2(m + l - 2) ≥ p2- q2, then ξ cannot be represented by a smoothly embedded 2-sphere. If 2(m + r + [ (l - r - 1)/4 ] - 1) ≥ p2- q2for 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 df≥ [ l - r - 1)/4 ] + 1.