Positive $2$-spheres in $4$-manifolds of signature $(1,n)$

Kazunori Kikuchi

Vol. 160 (1993), No. 2, 245–258

DOI: 10.2140/pjm.1993.160.245

Abstract

We sharpen Donaldson’s theorem on the standardness of definite intersection forms of smooth 4-manifolds in the same sense as Kervaire and Milnor sharpened Rohlin’s signature theorem. We then apply the result thus obtained to show that the homology classes of rational surfaces with $b_2^{-}\le 9$ which can be represented by smoothly embedded 2-spheres $S$ with $S\cdot S>0$ are up to diffeomorphism represented by smooth rational curves. Furthermore, we not only extend part of the application to the case where $b_2^{-}>9$, but also give an algorithm to see whether or not a given homology class of rational surfaces with $b_2^{-}\le 9$ can be represented by a smoothly embedded 2-sphere.

Mathematical Subject Classification

Primary: 57R95

Secondary: 57R19

Milestones

Received: 5 February 1992

Published: 1 October 1993