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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 positive homology classes of $ {\bf C}{\rm P}\sp 2\char93 2\overline{{\bf C}{\rm P}}{}\sp 2$ and $ {\bf C}{\rm P}\sp 2\char93 3\overline{{\bf C}{\rm P}}{}\sp 2$


Author: Kazunori Kikuchi
Journal: Proc. Amer. Math. Soc. 117 (1993), 861-869
MSC: Primary 57R95; Secondary 57N13
MathSciNet review: 1131036
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Abstract: Theorems of Donaldson are used to give a necessary and sufficient condition for a given second integral homology class $ \xi $ of $ {\mathbf{C}}{P^2}\char93 n{\overline {{\mathbf{C}}P} ^2}$ to be represented by a smoothly embedded $ 2$-sphere (1) for $ n = 2,\;3$ and $ \xi $ positive (with self-intersection positive), and (2) for $ n = 3$ and $ \xi $ characteristic. Case (2) is a consequence of a more general result on the characteristic embedding of $ 2$-spheres into $ 4$-manifolds, which result generalizes the theorem of Donaldson on spin $ 4$-manifolds just as the result of Kervaire and Milnor on the characteristic embedding extends Rohlin's signature theorem.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1131036-X
PII: S 0002-9939(1993)1131036-X
Keywords: Representing homology classes, almost definite $ 4$-manifold, orthogonal group, 11/8 conjecture
Article copyright: © Copyright 1993 American Mathematical Society