Representing characteristic homology classes of 𝑚𝐂𝐏²#𝐧\overline{𝐂}𝐏²

Guo, Jian Han; Gan, Dan Yan

doi:10.1090/S0002-9939-1994-1205494-7

Representing characteristic homology classes of $m\mathbf {C}\mathrm {P}^2 \# n\mathbf {\overline {C}}\mathrm {P}^2$
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by Jian Han Guo and Dan Yan Gan PDF

Proc. Amer. Math. Soc. 121 (1994), 1251-1255 Request permission

Abstract:

We prove the following theorems. Theorem 1. If $m,n \geq 1,x \in {H_2}(mC{P^2}\# n{\overline {CP} ^2})$ is a characteristic homology class with ${x^2} = 16l + m - n > 0$ and (1) $m < 3l + 1$ provided $l \geq 0$, or (2) $m < - 19l + 1$ provided $l < 0$. Suppose that the 11/8-conjecture is true. Then x cannot be represented by a smoothly embedded 2-sphere. Theorem 2. Let $m,n \geq 4l > 0,x \in {H_2}(mC{P^2}\# n{\overline {CP} ^2})$ be a primitive characteristic homology class with ${x^2} = \pm 16l + m - n$. Then x can be represented by a smoothly embedded 2-sphere.

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