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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Representing characteristic homology classes of $ m\bold C{\rm P}\sp 2\,\char93 \,n\overline {{\bf C}{\rm P}}{}\sp 2$


Authors: Jian Han Guo and Dan Yan Gan
Journal: Proc. Amer. Math. Soc. 121 (1994), 1251-1255
MSC: Primary 57R95; Secondary 57R40
MathSciNet review: 1205494
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Abstract: We prove the following theorems.

Theorem 1. If $ m,n \geq 1,x \in {H_2}(mC{P^2}\char93 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}\char93 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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1205494-7
PII: S 0002-9939(1994)1205494-7
Keywords: Representing, characteristic homology class, primitive
Article copyright: © Copyright 1994 American Mathematical Society