  We construct instanton Floer homology for lens spaces $L(p,q)$. As an
application, we prove that $X = \CP^2 # \CP^2$ does not admit a decomposition
$X = X_1 \cup X_2$. Here $X_1$ and $X_2$ are oriented, simply connected,
non-spin 4-manifolds with $b^+ = 1$ and with boundary $L(p, 2)$, and $p$ is a
prime number of the form $16N+1$.
