Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57R95, 57N13

Retrieve articles in all journals with MSC: 57R95, 57N13

Additional Information

Keywords: Representing homology classes, almost definite $ 4$-manifold, orthogonal group, 11/8 conjecture
Article copyright: © Copyright 1993 American Mathematical Society