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Theorems of Donaldson are used to give a necessary and sufficient condition for a given second integral homology class      ξ  \xi       of      C P 2 #  n C P ¯  2 {\mathbf {C}}{P^2}\# n{\overline {{\mathbf {C}}P} ^2}       to be represented by a smoothly embedded      2 2      -sphere (1) for      n = 2 , 3 n = 2,\;3       and      ξ  \xi       positive (with self-intersection positive), and (2) for      n = 3 n = 3       and      ξ  \xi       characteristic. Case (2) is a consequence of a more general result on the characteristic embedding of      2 2      -spheres into      4 4      -manifolds, which result generalizes the theorem of Donaldson on spin      4 4      -manifolds just as the result of Kervaire and Milnor on the characteristic embedding extends Rohlin’s signature theorem.

Representing positive homology classes of 𝐶𝑃²#2\overline{𝐶𝑃}² and 𝐶𝑃²#3\overline{𝐶𝑃}²