Representing positive homology classes of $\textbf {C}\textrm {P}^ 2\#2\overline {\textbf {C}\textrm {P}}{}^ 2$ and $\textbf {C}\textrm {P}^ 2\#3\overline {\textbf {C}\textrm {P}}{}^ 2$

Abstract:

Theorems of Donaldson are used to give a necessary and sufficient condition for a given second integral homology class $\xi$ of ${\mathbf {C}}{P^2}\# n{\overline {{\mathbf {C}}P} ^2}$ 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.